978-3-98547-561-2-volume-3

INTERNATIONAL GONGRESS

> L

is an invertible sheaf on

A

, then we have the Weil height

h_{A, L}

from Section 2.1. Recall that

h_{A, L}

is only defined up to addition of a bounded function on

A (\bar{Q})

. For abelian varieties, there is a canonical choice of function in the equivalence class

h_{A, L}

called the canonical or NÃ©ron-Tate height. A general reference for this section is [9, CHAPTER 9].

For an integer

n \in Z

, let

[n]

denote the multiplication-by-

n

endomorphism of

A

. Then

L

is called even or symmetric if there is an isomorphism

[- 1]^{*} L ≅ L

. It is called odd or antisymmetric if

[- 1]^{*} L ≅ L \otimes (- 1)

. If

L

is any ample invertible sheaf on

A

, then

L \otimes [- 1]^{*} L

is ample and even. So any abelian variety admits an even, ample invertible sheaf.

Suppose that

L

is even. Then [2]*

L ≅ L^{\otimes 4}

is a consequence of the Theorem of the Cube. So Theorem 2.4 implies

h_{A, L} \circ [2] = 4 h_{A, L}

as classes and by iteration

h_{A, L} \circ [2^{k}] =

4^{k} h_{A, L}

for all

k \geq 1

. We fix a representative

h_{A, L}^{'}

h_{A, L}

and find

h_{A, L}^{'} \circ [2^{k}] =

4^{k} h_{A, L}^{'} + O_{k} (1)

A (\bar{Q})

. Tate's Limit Argument is used to show convergence in the following definition.

Definition 2.6. Let

L

be an even invertible sheaf on

A

and let

P \in A (\bar{Q})

. Then the limit

\begin{matrix} (2.3) & {\hat{h}}_{A, L} (P) = lim_{k \to \infty} \frac{h_{A, L}^{'} (P)}{4^{k}} \end{matrix}

exists and is independent of the choice of representative

h_{A, L}^{'}

h_{A, L}

. The real-valued function

P \mapsto {\hat{h}}_{A, L} (P)

is called the canonical or NÃ©ron-Tate height (on

A

attached to

L

L

is even, then (2.3) immediately implies

{\hat{h}}_{A, L} ([2] (P)) = 4 {\hat{h}}_{A, L} (P)

for all

P \in A (\bar{Q})

. If

P

has finite order, then

[2^{m}] (P) = [2^{n}] (P)

for distinct integers

0 \leq m < n

by the Pigeonhole Principle. Thus

{\hat{h}}_{A, L} (P) = 0

There is nothing special about [2]. Indeed, one can replace [2] by

[m

] in (2.3) for any integer

m \geq 2

; one then needs to replace

4^{k}

in the denominator by

m^{2 k}

What happens if

L

is an odd invertible sheaf? In this case, [2]

^{*} L ≅ L^{\otimes 2}

. Then a similar limit (2.3) exists, but now we need to divide by

2^{k}

The set of odd invertible sheaves is a divisible subgroup of

Pic (A)

. From this, one can show that, after possibly extending the base field

F

, any invertible sheaf

L

A

decom-
poses as

L_{+} \otimes L_{-}

with

L_{+}

even and

L

- odd. One then defines

{\hat{h}}_{A, L} = {\hat{h}}_{A, L_{+}} + {\hat{h}}_{A, L}

; the decomposition of

L

is not quite unique, but this ambiguity does not affect

{\hat{h}}_{A, L}

For our purposes, we often restrict to even invertible sheaves.

Let us collect the some important facts about the NÃ©ron-Tate height.

Theorem 2.7. Let us keep the notation above. In particular,

A

is an abelian variety defined over a number field

F \subseteq \bar{Q}

(i) Then association

L \mapsto {\hat{h}}_{A, L}

is a group homomorphism from

Pic (V)

to the additive group of real-valued maps

A (\bar{Q}) \to R

Suppose

L

is an invertible sheaf on

A

(ii) The NÃ©ron-Tate height

{\hat{h}}_{A, L}

represents the Weil height

h_{A, L}

(iii) If

L

is even, then the parallelogram equality

{\hat{h}}_{A, L} (P + Q) + {\hat{h}}_{A, L} (P - Q) = 2 {\hat{h}}_{A, L} (P) + 2 {\hat{h}}_{A, L} (Q)

holds for all

P, Q \in A (\bar{Q})

(iv) If

L

is even and ample, then

{\hat{h}}_{A, L}

takes nonnegative values and vanishes precisely on

A_{tors}

(v) If

L

is even and ample, then

{\hat{h}}_{A, L}

induces a well-defined map

A (\bar{Q}) \otimes R \to

[0, \infty)

. It is the square of a norm

∥ \cdot ∥

on the

R

-vector space

A (\bar{Q}) \otimes R

and satisfies the parallelogram equality.

The norm

∥ \cdot ∥

allows us to do geometry in the

R

-vector space

A (\bar{Q}) \otimes R

(which is infinite dimensional if

\dim A \geq 1)

. Indeed, for

z, w \in A (\bar{Q}) \otimes R

, we define

⟨ P, Q ⟩ = \frac{1}{2} (∥ P + Q ∥^{2} - ∥ P ∥^{2} - ∥ Q ∥^{2})

Then

⟨ \cdot, \cdot ⟩

is a positive definite, symmetric, bilinear form.

By abuse of notation, we also write

∥ P ∥

and

⟨ P, Q ⟩

for

P, Q \in A (\bar{Q})

. In this notation we have

⟨ P, P ⟩ = {\hat{h}}_{A, L} (P)

The Mordell-Weil Theorem implies that

A (F) \otimes R

is finite dimensional. We will see that

∥ \cdot ∥

is a suitable norm to do Euclidean geometry in

A (F) \otimes R

3. VOJTA'S APPROACH TO THE MORDELL CONJECTURE

Recall that the Mordell Conjecture was proved first by Faltings. In this section we briefly describe Vojta's approach to the Mordell Conjecture [62]. At the core is the deep Vojta inequality which we state here for a curve in an abelian variety.

Let

A

be an abelian variety defined over a number field

F \subseteq \bar{Q}

. Let

L

be an ample and even invertible sheaf on

A

. We write

∥ \cdot ∥ = {\hat{h}}_{A, L}^{1 / 2}

for the norm on

A (\bar{Q}) \otimes R

defined in Theorem 2.7.

Theorem 3.1 (Vojta's inequality). Let

C \subseteq A

be a curve that is defined over

F

and that is not a translate of an algebraic subgroup of A. There are

c_{1} > 1, c_{2} > 1

, and

c_{3} > 0

with the following property. If

P, Q \in C (\bar{Q})

satisfy

⟨ P, Q ⟩ \geq (1 - \frac{1}{c_{1}}) ∥ P ∥ ∥ Q ∥

and

∥ Q ∥ \geq c_{2} ∥ P ∥

then

∥ P ∥ \leq c_{3}

We refer also to RÃ©mond's work [55] for a completely explicit version of Vojta's inequality.

The values

c_{1}, c_{2}, c_{3}

depend on the curve

C

. One remarkable aspect is that Vojta's inequality is a statement about pairs of

\bar{Q}

-points of the curve

C

. So all three values

c_{1}, c_{2}, c_{3}

are "absolute," i.e., we can take them as independent of the base field

F

A

and

C

. Both

c_{1}

and

c_{2}

are of "geometric nature." They depend only on the degree of

C

with respect to

L

and other discrete data attached to

A

and

C

. In contrast,

c_{3}

is of "arithmetic nature." Roughly speaking, it depends on suitable heights of coefficients that define the curve

C

in some projective embedding.

Let us now sketch a proof of Mordell's Conjecture using the Vojta inequality and the classical Mordell-Weil Theorem.

Suppose

C

has genus

g \geq 2

. Without loss of generality,

C (F) \neq \emptyset

. So we fix a base point

P_{0} \in C (F)

, then

P \mapsto P - P_{0}

induces an immersion

C \to Jac (C)

. So we may assume that

C

is a curve inside the

g

-dimensional

A = Jac (C)

. Note that

C

is not a translate of an algebraic subgroup of its Jacobian since

g \geq 2

We observe that

C (F) = C (\bar{Q}) \cap Jac (C) (F)

By the Northcott property, stated below Theorem 2.4, combined with Theorem 2.7(ii) we find that the "ball"

\begin{matrix} (3.1) & {P \in C (F) : ∥ P ∥^{2} \leq B} \end{matrix}

is finite for all

B

We split the set of points

C (F)

into two subsets:

\begin{array}{ll} {P \in C (F) : ∥ P ∥^{2} > c_{3}} & (large points), \\ {P \in C (F) : ∥ P ∥^{2} \leq c_{3}} & (moderate points). \end{array}

By the finiteness statement around (3.1), it suffices to show that there are at most finitely many large points.

For any

z \in Jac (C) (F) \otimes R

, we define the truncated cone

\begin{aligned} T (z) & = {w \in Jac (C) (F) \otimes R : ⟨ z, w ⟩ \geq (1 - 1 / c_{1}) ∥ z ∥ ∥ w ∥ and ∥ w ∥^{2} > c_{3}} \\ \subseteq Jac (C) (F) \otimes R \end{aligned}

By the Mordell-Weil Theorem,

Jac (C) (F) \otimes R

is a finite-dimensional

R

-vector space. So the unit sphere with respect to the norm

∥ \cdot ∥

coming from the NÃ©ron-Tate height
is compact. Therefore,

{w \in Jac (C) (F) \otimes R : ∥ w ∥^{2} > c_{3}}

is covered by a finite union

T (z_{1}) \cup \dots \cup T (z_{N})

. Using a sphere packing argument, one can arrange that

N

is bounded from above by

c^{'} \cdot c^{r k J a c (C) (F)}

where

c^{'} > 0

and

c > 1

depend only on

c_{1}

. This observation will be important for deriving uniform bounds for

# C (F)

Any large point in

C (F)

has image in some

T (z_{j})

from above. After possibly adjusting

N

, one can arrange that each

z_{j}

is the image of a point

P_{j} \in C (F)

with

{∥ P_{j} ∥}^{2} > c_{3}

for all

j \in {1, \dots, N}

. If

Q \in C (F)

has image in

T (z_{j})

, then Vojta's inequality implies

{\hat{h}}_{L} (Q)^{1 / 2} = ∥ Q ∥ \leq c_{2} ∥ P_{j} ∥

. But then

Q \in C (F)

lies in a finite ball as in (3.1). So the number of possible

Q

that come to lie in a single

T (z_{j})

is finite. Thus

C (F)

is finite.

The constants

c_{1}, c_{2}

, and

c_{3}

in Vojta's inequality can be made effective in terms of

A

and

C

. Yet, the proof as a whole is ineffective. Indeed, the height bound for

Q

depends on the hypothetical point

P_{j}

. However, there is no guarantee that

P_{j}

exists and if it does not, there is no known way to know for sure.

Using Mumford's Gap Principle, one can show that the number of large points

C (F)

that come to lie in a single

T (z_{j})

is bounded from above by

c^{'} \cdot c^{r k J a c (C) (F)}

, after possibly increasing the constants. Now we need to introduce dependency on

c_{2}

. But the base

c

will remain geometric in nature, it depends on the genus of

g

. But it does not depend on

c_{3}

or other arithmetic properties of

C

that encode the heights of coefficients defining the said curve. Finally, as observed by Bombieri, 7 is admissible for

c

for any genus. Indeed, he showed that 4 is admissible for

c_{1}

Recall that Vojta's inequality with the same values of

c_{1}, c_{2}, c_{3}

applies to points in

C (F^{'})

for all finite extensions

F^{'} / F

. The upshot is that the number of large points of

C (F^{'})

is bounded by

c^{'} \cdot c^{r k J a c (C) (F^{'})}

where

c, c^{'}

depend on

C

, but not on

F^{'}

The dichotomy between large and moderate points was already visible in Vojta's work. But its origin is older and already appears in modified form in work of Thue, Siegel, Mahler, and Roth on diophantine approximation.

RÃ©mond's explicit ThÃ©orÃ¨me 2.1 [54] gives a recipe how to bound the total number of rational points using a bound for the number of moderate points.

With our eyes set on Mazur's question, we aim to obtain good bounds for the number of moderate points. In the coming two sections we explain our general approach to the proof of Theorem 1.12.

4. COMPARING WEIL AND NÃ‰RON-TATE HEIGHTS

The interplay between the Weil and NÃ©ron-Tate heights on a family of abelian varieties leads to powerful results including Silverman's Specialization Theorem [57] and more recent work by Masser and Zannier towards the relative Manin-Mumford Conjecture [44]. This interaction also plays a central role in the proof of Theorem 1.12 that resolved Mazur's question.

Having worked with a fixed abelian variety in Sections 2.2 and 3, we now shift gears and work in a family of abelian varieties.

Example 4.1. Let

Y (2) = P^{1} ∖ {0, 1, \infty}

. For

λ \in Y (2) (C)

, we have an elliptic curve

E_{λ} \subseteq P^{2}

determined by

y^{2} z = x (x - z) (x - λ z)

where the origin is

[0 : 1 : 0]

. The total space

E

is a surface presented with a closed immersion

E ↪ P^{2} \times Y (2)

. It is called the Legendre family of elliptic curves and is an abelian scheme over

Y (2)

. So we can add two complex points of

E

if they are in the same fiber above

Y (2)

. More precisely, there is an addition morphism

E \times_{S} E \to E

over

S

, as well as an inversion morphism

E \to E

over

S

. Finally, the zero section of

E

is given by

λ \mapsto ([0 : 1 : 0], λ)

Consider a geometrically irreducible smooth quasiprojective variety

S

defined over a number field

F \subseteq \bar{Q}

. Let

π : A \to S

be an abelian scheme over

S

. So each fiber

A_{s} = π^{- 1} (s)

, where

s \in S (\bar{Q})

, is an abelian variety. We have an addition morphism on the fibered square

A \times \times_{S} A \to A

and an inversion morphism

A \to A

; both are relative over

S

. Addition induces a multiplication-by-

n

morphism

[n] : A \to A

over

S

for all

n \in Z

For simplicity, we assume that

A

is presented with an immersion

A ↪ P^{n} \times S

over

S

, much as in Example 4.1 above. Let

L

be the restriction of the hyperplane bundle

O (1)

P^{n} \times S = P_{S}^{n}

A

. We also assume that

L

is even, that is

[- 1]^{*} L ≅ L

. This allows us to define a fiberwise NÃ©ron-Tate height on

A (\bar{Q})

which we abbreviate by

{\hat{h}}_{A}

Let

s \in S (\bar{Q})

. Then

A_{s}

is an abelian variety in

P^{n}

. We have two functions,

{{\hat{h}}_{A} |}_{A_{s} (\bar{Q})}

and

{h |}_{A_{s} (\bar{Q})}

; the latter is the restriction of the Weil height on

P^{n}

. By Theorem 2.7(ii), their difference is bounded in absolute value in function of

s

In the example of the Legendre family, the point

[λ : 0 : 1] \in E_{λ}

is of order 2 for all

λ

. So its NÃ©ron-Tate height vanishes, but its Weil height equals

h ([λ : 1])

and is thus unbounded as

λ

varies.

We would like to understand the difference between NÃ©ron-Tate and Weil heights on

A

as the base point

s \in S (\bar{Q})

varies. As suggested by the Legendre case, the key is the Weil height on the base

S

. To keep things concrete, we will assume that

S

comes with an immersion

S ↪ P^{m}

. We identify

S

with a Zariski locally closed subset of

P^{m}

. So

S

need not be projective, but its Zariski closure

\bar{S}

P^{m}

is. We write

h_{S}

for

{h |}_{\bar{S} (\bar{Q})} : \bar{S} (\bar{Q}) \to [0, \infty)

where

h

is the Weil height on

P^{m} (\bar{Q})

. In the language of Section 2.1,

h_{S}

represents the Weil height attached to

(\bar{S}, {O (1) |}_{\bar{S}})

The difference between Weil and NÃ©ron-Tate heights on the total space

A (\bar{Q})

was clarified in work of Zimmer [73] in the elliptic setting and Manin-Zarhin [69] and SilvermanTate [57] in the more general setting. In our case the latter result amounts to

for all

s \in S (\bar{Q})

We introduce a final player, a geometrically-irreducible subvariety

V

A

defined over

F

Theorem 4.2 (Silverman [57]). Suppose

S

and

V \subseteq A

are curves such that

V

dominates

S

. Then

\begin{matrix} (4.2) & lim_{\begin{matrix} P \in V (\bar{Q}) \\ h (π (P)) \to \infty \end{matrix}} \frac{{\hat{h}}_{A} (P)}{h (π (P))} \end{matrix}

exists. Suppose, in addition, that the geometric generic fiber of

A \to S

has trivial trace over

\bar{Q}

. Then the limit vanishes if and only if

V

is an irreducible component of

\ker [N]

for some

N \geq 1

. Otherwise the limit is positive.

Silverman computed the limit in terms of the NÃ©ron-Tate height of

V

restricted to the generic fiber

A \to S

The "if direction" is straightforward: in this case all

P

in question are of finite order and their NÃ©ron-Tate height vanishes; see Theorem 2.7(iv). The "only if" direction is deeper and has many applications: Silverman's Specialization Theorem, Theorem C in [57], as well as applications to unlikely intersections by Masser and Zannier, see [44] and [68] for an overview and more results.

What happens if

V

has dimension

> 1

and

S

remains a curve? In this case the limit (4.2) does not make sense. Indeed, for fixed

s \in S (\bar{Q})

, the set of

P \in V (\bar{Q})

that map to

s

has positive dimension and thus unbounded NÃ©ron-Tate height.

Motivated by Theorem 4.2, the author showed the next theorem. It may serve as a higher-dimensional substitute for Silverman's Theorem 4.2. For an irreducible subvariety of

V

A

that dominates

S

, we write

V_{\bar{η}}

for the geometric generic fiber of

{π |}_{V} : V \to S

. This is a possibly reducible subvariety of the geometric generic fiber

A_{\bar{η}}

A \to S

Theorem 4.3 ([35]). Suppose

S = Y (2)

and let

A = E [g]

be the

g

-fold fibered power of the Legendre family of elliptic curves. Suppose

V \subseteq E^{[g]}

dominates

Y (2)

and

V_{\bar{η}}

is not a finite union of irreducible components of algebraic subgroups of

A_{\bar{η}}

Then there exist

c (V) > 0

and a Zariski open and dense subset

U \subseteq V

with

\begin{matrix} (4.4) & h_{Y (2)} (π (P)) \leq c (V) max {1, {\hat{h}}_{A} (P)} for all P \in U (\bar{Q}) \end{matrix}

Say (4.3) holds. If

P \in U (\bar{Q})

has finite order as a point in its respective fiber, we find

h_{Y (2)} (π (p)) \leq c (V)

and the total Weil height of

P

is bounded from above by (4.1). This simple observation led to the resolution of several "special points" problems [35] in the spirit of the AndrÃ©-Oort Conjecture. For example, torsion points

P \in V (\bar{Q})

that lie in a fiber with complex multiplication are not Zariski dense in

V

. The proof of Theorem 4.3 makes use of Siu's Criterion, see Remark 2.5(iii), and an investigation of monodromy in

E^{[g]}

The Zariski open

U

cannot in general be taken to equal

V

. But there is a natural description of this set in geometric terms through unlikely intersections.

The hypothesis (4.3) is necessary and essentially rules out that

V

itself is a family of abelian subvarieties.

Gao and the author [33] then generalized Theorem 4.3 to an abelian scheme when the base is again a smooth curve

S

defined over

\bar{Q}

. Here more care is needed in connection
with the hypothesis (4.3). Indeed, if

A = A \times S

is a constant abelian scheme, where

A

is an abelian variety, then (4.4) cannot hold generically for

V = Y \times S

. Roughly speaking, the condition in [33] that replaces (4.3) also needs to take into account a possible constant part of

A_{\bar{η}}

. If

A_{\bar{η}}

has no constant part, i.e., if its

\bar{Q} (\bar{η}) / \bar{Q}

-trace is 0 , then (4.3) suffices for

S

a curve. The case of a higher-dimensional base requires even more care, as we will see.

There were two applications of the height bound in [33].

First, and in the same paper, we proved new cases of the geometric Bogomolov Conjecture for an abelian variety defined over the function field of the curve

S

. This approach relied on Silverman's Theorem 4.2. It was used earlier in [35] to give a new proof of the Geometric Bogomolov Conjecture in a power of an elliptic curve. The number field case of the Bogomolov Conjecture was proved by Ullmo [61] and Zhang [71] in the 1990s. Progress in the function field case was later made by Cinkir, Faber, Moriwaki, Gubler, and Yamaki. For the state of the Geometric Bogomolov Conjecture as of 2017, we refer to a survey of Yamaki [64]. Gubler's strategy works in arbitrary characteristic and was expanded on by Yamaki. In joint work [10] with Cantat, Gao, and Xie, the author later established the Geometric Bogomolov Conjecture in characteristic 0 by bypassing the height inequality (4.4). Very recently, Xie and Yuan [63] announced a proof of the Geometric Bogomolov Conjecture in arbitrary characteristic. Their approach builds on the work of Gubler and Yamaki.

Second, and in later joint work with Dimitrov and Gao [23], we established uniformity for the number of rational points in the spirit of Mazur's question for curves parametrized by the 1-dimensional base

S

As we shall see, the proof of Theorem 1.12 requires a height comparison result like (4.4) for abelian schemes over a base

S

of any dimension. But now the correct condition to impose on

V

is more sophisticated and cannot be easily read off of the geometric generic fiber as in (4.3). The condition relies on the Betti map, which we introduce in the next section.

4.1. Degenerate subvarieties and the Betti map

In this section,

S

is a smooth irreducible quasiprojective variety over

C

. Let

π : A \to S

again be an abelian scheme over

S

of relative dimension

g \geq 1

For each

s \in S (C)

, the fiber

A_{s} (C)

is a complex torus of dimension

g

. Forgetting the complex structure, each

g

-dimensional complex torus is diffeomorphic to

(R / Z)^{2 g}

as a real Lie group. By Ehresmann's Theorem, this diffeomorphism extends locally in the analytic topology on the base. That is, there is a contractible open neighborhood

U

s

S (C)

and a diffeomorphism

A_{U} = π^{- 1} (U) \to (R / Z)^{2 g} \times U

over

U

. Fiberwise this diffeomorphism can be arranged to be a group isomorphism above each point of

U

. Thus we can locally trivialize the abelian scheme at the cost of sacrificing the complex-analytic structure.

The trivialization is not entirely unique as we can let a matrix in

{G L}_{2 g} (Z)

act in the natural way on the real torus

(R / Z)^{2 g}

. But since

U

is connected, this is the only ambiguity. It is harmless for what follows.

The Betti map

β_{U}

attached to

U

is the composition of the trivialization followed by the projection

This map has appeared implicitly in diophantine geometry in work of Masser and Zannier [44]. We also refer to more recent work of AndrÃ©, Corvaja, and Zannier [3] for a systematic study of the Betti map.

We list some of the most important properties:

(i) For all

s \in U

, the restriction

{β_{U} |}_{A_{s} (C)} : A_{s} (C) \to (R / Z)^{2 g}

is a diffeomorphism of real Lie groups. In particular,

P \in A_{U}

has finite order in its respective fiber if and only if

β_{U} (P) \in (Q / Z)^{2 g}

(ii) For all

P \in U

the fiber

β_{U}^{- 1} (β_{U} (P))

is a complex-analytic subset of

A_{U}

Definition 4.4. An irreducible closed subvariety

V \subseteq A

that dominates

S

is called degenerate if for all

U

and

β_{U}

as above and all smooth points

P

V_{U} = {π |}_{V}^{- 1} (U)

the differential of

d_{P} ({β_{U} |}_{V_{U}})

satisfies

\begin{matrix} (4.5) & {rkd}_{P} ({β_{U} |}_{V_{U}}) < 2 \dim V \end{matrix}

It has become customary to call

V

degenerate if it is not nondegenerate.

For all smooth points

P

V_{U}

, the left-hand side of (4.5) is at most the right-hand side, which equals the real dimension of

V_{U}

. It is also at most

2 g

, the real dimension of a fiber of

A \to S

. Moreover, if the maximal rank of

d β_{U}

V_{U}

is attained at

P

then the maximal rank is attained also in a neighborhood of

P

V_{U}

. Being nondegenerate is a local property.

Let us consider some examples.

Example 4.5. (i) If

S

is a point, then

A

is an abelian variety and an arbitrary subvariety

V \subseteq A

is nondegenerate because

β_{S}

is a diffeomorphism.

(ii) Suppose

\dim V > g

. Then

{rkd}_{P} (β_{U} ∣ V_{U}) \leq 2 g < 2 \dim V

for all smooth

P

and so

V

is degenerate. In particular,

A

is a degenerate subvariety of

A

\dim S \geq 1

(iii) Suppose

A = A \times S

is a constant abelian scheme with

A

an abelian variety. If

Y \subseteq A

is a closed irreducible subvariety and if

\dim S \geq 1

, then

Y \times S

is degenerate. Indeed, the rank is at most

2 \dim Y < 2 \dim Y \times S

(iv) Suppose

V

is an irreducible component of

\ker [N]

for some integer

N \geq 1

. Any point in

V (C)

has order dividing

N

(and, in fact, equal to

N

). So the image of

{β_{U} |}_{V_{U}}

is finite and hence

V

is degenerate if

\dim S \geq 1

(v) Suppose

V

is the image of a section

S \to A

. If the geometric generic fiber of

A \to S

has trivial trace, then

{(β_{U}) |}_{V_{U}}

is constant if and only if

V

is an irreducible component of

\ker [N]

for some

N \geq 1

. This is Manin's Theorem of the Kernel, we refer to Bertrand's article [7] for the history of this theorem.
(vi) Suppose

A = E^{[g]}

and

V

are as in Theorem 4.3. One step in the proof of this theorem consisted in verifying that

V

, subject to hypothesis (4.3), is nondegenerate. Crucial input came from the monodromy action of the fundamental group of the base

Y (2) = P^{1} ∖ {0, 1, \infty}

on the first homology of a fiber

A_{s}

with

s

in general position. In this case the monodromy action is unipotent at the cusps 0 and 1 of

Y

(2). This enabled the author to use a result of Kronecker from diophantine approximation. Already Masser and Zannier [44] used the monodromy action in their earlier work for

V

a curve.

(vii) If

S

is a curve, then the monodromy action of the fundamental group of

S (C)

on the homology of fibers of

A \to S

is locally quasiunipotent. But if

S

is projective, then there are no cusps. So exploiting monodromy in this setting required a different approach. In [33] Gao and the author used o-minimal geometry and the Pila-Wilkie Counting Theorem [51]. A related case was solved by Cantat, Gao, and Xie in collaboration with the author [10]; we used dynamical methods.

(viii) Finally, we consider the case of an abelian scheme

A

over a base

S

of arbitrary dimension. This setting was studied recently in work of AndrÃ©Corvaja-Zannier [3]. Moreover, the work of Gao on the Ax-Schanuel Theorem [30] for the universal family of abelian varieties led him to formulate a geometric condition [29] that guarantees nondegeneracy. It proves crucial in the application to Mazur's question and we will return to this point. Gao's result also relied on o-minimal geometry and the Pila-Wilkie Theorem.

4.2. Comparing the Weil and NÃ©ron-Tate heights on a subvariety

We now come to the generalization of Theorem 4.3 to nondegenerate subvarieties. We retain the notation introduced in Section 4.1. So

S

is a smooth irreducible quasiprojective variety defined over

\bar{Q}

equipped with an immersion in

P^{m}

. We have a height

h_{S}

\bar{S} (\bar{Q})

. Moreover,

π : A \to S

is an abelian scheme over

S

presented with an immersion

A \to P^{n} \times S

over

S

. Finally,

L

is as in Section 4 and

{\hat{h}}_{A}

is the fiberwise NÃ©ron-Tate height on

A (\bar{Q})

We assume that

A

carries symplectic level-

ℓ

structure for some fixed

ℓ \geq 3

and that

L

induces a principal polarization. For the proof of Theorem 1.12, it suffices to have the following height bound under these conditions. We also refer to [24, THEOREM B.1] for a version that relaxes some of the conditions.

Theorem 4.6 ([24, THEOREM 1.6]). Let

V

be a nondegenerate irreducible subvariety of

A

that dominates

S

. There exist

c (V) > 0, c^{'} (V) \geq 0

, and a Zariski open and dense subset

U \subseteq V

with

h_{S} (π (P)) \leq c (V) {\hat{h}}_{A} (P) + c^{'} (V) for all P \in U (\bar{Q})

We refer to Yuan and Zhang's Theorem 6.2.2 [67] for a height inequality in the dynamical setting.

Here are just a few words on the proof of Theorem 4.6. Siu's Criterion, see Remark 2.5, is used to compare the Weil height of

π (P)

with a Weil height of

P

. The nondegeneracy hypothesis is used to extract a volume estimate. The upshot is a lower bound for the top self-intersection number in Siu's Criterion. The predecessor of Theorem 4.6 in the earlier works

[33, 35]

was proved by counting torsion points using the Geometry of Number; volumes played an important role here as well. Passing from the Weil to the NÃ©ron-Tate height introduces an additional dependency on the height of

π (P)

, see (4.1). However, this contribution can be eliminated by using Masser's "ruthless strategy of killing Zimmer constants" [68, APPENDIX c]. This task is done by repeated iteration of the duplication morphism [2] which has the effect of truncating Tate's Limit Process after finitely many steps. Our ambient group scheme

A

is quasiprojective but in general not projective. So a suitable compactification is required that admits some compatibility with the duplication morphism.

The positive constant

c (V)

in Theorem 4.6 ultimately comes from the application of Siu's Criterion. As such it can expressed in geometric terms.

5. APPLICATION TO MODERATE POINTS ON CURVES

In this section we sketch the main lines of the proof of Theorem 1.12. It will be enough to bound the number of moderate points, see Section 3.

5.1. The Faltings-Zhang morphism

Smooth curves of genus

g \geq 2

defined over

\bar{Q}

are classified by the

\bar{Q}

-points of a quasiprojective variety, the coarse moduli space. For us it is convenient to work with symplectic level-

ℓ

structure on the Jacobian for some fixed integer

ℓ \geq 3

. With this extra data, we obtain a fine moduli space

M_{g}

, together with a universal family

C_{g} \to M_{g}

. Fibers of this family are smooth curves of genus

g

with the said level structure on the Jacobian. Then

M_{g}

carries the structure of a smooth quasiprojective variety of dimension

3 g - 3

defined over a cyclotomic field. For convenience, we replace

M_{g}

by an irreducible component by choosing a complex root of unity of order

ℓ

and consider it as defined over

\bar{Q}

The Torelli morphism

τ : M_{g} \to A_{g}

takes a smooth curve to its Jacobian with the level structure; here

A_{g}

denotes the fine moduli space of

g

-dimensional abelian varieties with a principal polarization and symplectic level

- ℓ

structure.

Let

M \geq 0

be an integer and consider

M + 1

points

P_{0}, \dots, P_{M} \in C_{g} (C)

in the same fiber

C

C_{g} \to M_{g}

. The differences

[P_{1}] - [P_{0}], \dots, [P_{M}] - [P_{0}]

are divisors of degree 0 on

C

. We obtain

M

complex points in the Jacobian of

C

and so

M

complex points of

V_{g}

. We obtain a commutative diagram

of morphisms of schemes; here the exponent

[M]

denotes taking the

M

th fibered power over the base. The morphism

D

is called the Faltings-Zhang morphism; see [26, LEMMA 4.1] and [71, LEMMA 3.1] for important applications to diophantine geometry of variants of this morphism. The morphism

D

is proper.

A modified version of this construction is also useful. Say

S \to M_{g}

is a quasifinite morphism with

S

an irreducible quasiprojective variety defined over

\bar{Q}

. We obtain a proper morphism

D : C_{g}^{[M + 1]} \times_{M_{g}} S \to V_{g}^{[M]} \times_{A_{g}} S

, again called Faltings-Zhang morphism.

Gao, using his Ax-Schanuel Theorem for the universal family

N_{g}

[30] and a characterization [28] of bialgebraic subvarieties of

ℜ_{g}

, obtained

Theorem 5.1 (Gao [29]). Let

S \to M_{g}

be as above, i.e., a quasifinite morphism from an irreducible quasiprojective variety

S

defined over

\bar{Q}

and

g \geq 2

. If

M \geq \dim M_{g} + 1 =

3 g - 2

, then

D (E_{g}^{[M + 1]} \times_{M_{g}} S)

is a nondegenerate subvariety of

I_{g}^{[M]} \times_{A_{g}} S

Mok, Pila, and Tsimerman [47] earlier proved an Ax-Schanuel Theorem for Shimura varieties. Gao's result [30] is a "mixed" version in the abelian setting. We refer to the survey [4] on recent developments in functional transcendence.

The hypothesis

g \geq 2

is crucial. The definition of the Faltings-Zhang morphism makes sense for

g = 1

. But it will be surjective and the image is degenerate expect in the (for our purposes uninteresting) case

\dim S = 0

We consider here for simplicity only the case

S = M_{g}

Using basic dimension theory, we see

\dim D (S_{g}^{[M + 1]}) \leq M + 1 + \dim M_{g}

. The image lies in the fibered power

V_{g}^{[M]}

where the relative dimension is

M g

. A necessary condition for

D (C_{g}^{[M + 1]})

to be nondegenerate is

\dim D (C_{g}^{[M + 1]}) \leq M g

, see Example 4.5(ii). This inequality follows from

\begin{matrix} (5.1) & M + 3 g - 2 = M + 1 + \dim M_{g} \leq M g \end{matrix}

M \leq 3

, the numerical condition (5.1) is not satisfied for any

g \geq 2

. For this reason, we cannot hope to work with the image of

C_{g} \times_{M_{g}} C_{g}

V g

by taking differences. Moreover, there seems to be no reasonable way to work with a single copy of

C_{g}

, where the relations between dimensions would be even worse. The numerical condition (5.1) is satisfied for all

M \geq 4

and all

g \geq 2

. Gao's Theorem implies that

M \geq 3 g - 2

is sufficient to guarantee nondegeneracy.

We can thus apply Theorem 4.6 to the image

D (C_{g}^{[M + 1]})

of the Faltings-Zhang morphism in

V_{g}^{[M]} \times_{A_{g}} M_{g}

. Let

M = 3 g - 2

, then

\begin{matrix} (5.2) & h_{M_{g}} (s) \leq c (g) ({\hat{h}}_{N_{g}} (P_{1} - P_{0}) + \dots + {\hat{h}}_{O_{g}} (P_{M} - P_{0})) + c^{'} (g) \end{matrix}

for all

(P_{0}, \dots, P_{M}) \in U (\bar{Q})

above

s \in M_{g} (\bar{Q})

where

U

is a Zariski open and dense subset of

D (C_{g}^{[M + 1]})

. The constants

c (g) > 0

and

c^{'} (g) \geq 0

depend on the various choices made regarding projective immersions of

M_{g}

and

{N l}_{g}

. Ultimately, they depend only on

g

once these choices have been made.

The Zariski open

U

cannot be replaced by

D (C_{g}^{[M + 1]})

. Indeed, the right-hand side of (5.2) vanishes on the diagonal

P_{0} = P_{1} = \dots = P_{M}

whereas the left-hand side is unbounded as

s

varies.

Let us shift back to using

∥ \cdot ∥

to denote the square root of the NÃ©ron-Tate height, see Section 2.2. Let us assume that

\begin{matrix} (5.3) & h_{M_{g}} (s) \geq 2 c^{'} (g) \end{matrix}

2 M = 6 g - 4

we find

\begin{matrix} (5.4) & h_{M_{g}} (s) \leq c (g) (6 g - 4) max_{1 \leq j \leq M} {∥ P_{j} - P_{0} ∥}^{2} for all (P_{0}, \dots, P_{M}) \in U (\bar{Q}) \end{matrix}

Morally, (5.4) states that among a (

3 g - 1)

-tuple of points on a curve of genus

g

in general position, there must be a pair that repels one another with respect to the norm

∥ \cdot ∥

. The squared distance of such a pair is larger than a positive multiple, depending only on

g

, of the modular height

h_{M_{g}} (s)

; this is the key to bounding the number of moderate points from Section 3.

As stated at the end of Section 4.2, the value

c (g)

can be expressed in terms of geometry properties of the image of

S_{g}^{[M + 1]}

under the Faltings-Zhang morphism.

Question 5.2. What is an admissible value for

c (g)

5.2. Bounding the number of moderate points-a sketch

Recall that, by the discussion at the end of Section 3, we need to bound the number of moderate points.

We retain the notation of Sections 3 and 5. The curve

C

from Section 3 can be equipped with suitable level structure over a field

F^{'} / F

with

[F^{'} : F]

bounded in terms of

g

. The rank of

Jac (C) (F^{'})

may be dangerously larger than the rank of

Jac (C) (F)

. But recall that we are interested in bounding

# C (F)

from above, so only the

group Jac (C) (F)

will be relevant. Moreover,

c_{1}, c_{2}

, and

c_{3}

from a suitable version of Vojta's inequality are unaffected by extending

F

. The effect is that we may identify

C

with a fiber of

C_{g}

above some point

s \in M_{g} (F^{'})

. For simplicity, we assume

F = F^{'}

for this proof sketch.

We require some additional information on

c_{3} (C)

. It turns out that we can take

c_{3} = c_{4} (g) max {1, h_{M_{g}} (s)}

where

c_{4} (g) > 0

depends on

g

. This follows RÃ©mond's work [55] on the Vojta inequality. A similar dependency is apparent in de Diego's result, Theorem 1.6.

Suppose now that (5.3) holds, so

h_{M_{g}} (s)

is sufficiently large in terms of

g

. We fix an auxiliary base point

P^{'} \in C (F)

. We must bound from above the number of points in

B (R) = {P \in C (F) : {∥ P - P^{'} ∥}^{2} \leq R^{2}} with R = {(c_{4} (g) h_{M_{g}} (s))}^{1 / 2}

where

∥ \cdot ∥

denotes the square root of the NÃ©ron-Tate height on the fiber of

N_{g} \to A_{g}

associated to the Jacobian of

C

Recall

M = 3 g - 2

and suppose

P_{0}, \dots, P_{M} \in C (F)

. If the tuple

(P_{0}, \dots, P_{M})

is in general position, i.e.,

(P_{1} - P_{0}, \dots, P_{M} - P_{0})

lies in

U (\bar{Q})

from (5.4), then there is

i

with

P_{i} \notin B (P_{0}, r) = {P \in C (F) : {∥ P - P_{0} ∥}^{2} \leq r^{2}} with r = {(c_{5} (g) h_{M_{g}} (s))}^{1 / 2}

If we had a guarantee that such

(M + 1)

-tuples of pairwise distinct points are always in general position, then

# B (P_{0}, r) < M = 3 g - 2

. By sphere packing, we can cover the image of

B (R)

Jac (C) (F) \otimes R

by at most

(1 + 2 R / r)^{r k J a c (C) (F)}

closed balls in

Jac (C) (F)

of radius

r

. One can even arrange for the ball centers to arise as points of

C (F)

. The modular height

h_{M_{g}} (s)

cancels out in the quotient

\frac{R}{r} = {(\frac{c_{4} (g)}{c_{5} (g)})}^{1 / 2}

This would complete the proof of Theorem 1.12 except that there is no reason to believe that

(P_{1} - P_{0}, \dots, P_{M} - P_{0}) \in U (\bar{Q})

(even if the

P_{j}

are pairwise distinct). Treating points with image in the complement of

U

requires induction on the dimension. Here we rely on the freedom to replace

M_{g}

by a subvariety in that Gao's Theorem 5.1.

Let us briefly explain the resulting induction step. Observe that the dimension of this exceptional set is at most

\dim D (C_{g}^{[M + 1]}) - 1 \leq M + \dim M_{g}

. There are two cases for

(P_{0}, \dots, P_{M})

with image in the exceptional set

(D (C_{g}^{[M + 1]}) ∖ U) (\bar{Q})

on which we do not have the height inequality. For the case study, recall that

s \in M_{g} (\bar{Q})

denotes the point below all the

P_{j}

and

τ (s) \in A_{g} (\bar{Q})

is its image under the Torelli morphism

τ

First, assume that the fiber of

D (C_{g}^{[M + 1]}) ∖ U \to A g

above

τ (s)

has dimension at most

M

. This fiber contains

(P_{1} - P_{0}, \dots, P_{M} - P_{0})

. This case is solved using a zero estimate motivated by the following simple lemma.

Lemma 5.3. Suppose

C

is an irreducible curve defined over

C

and

W

a proper Zariski closed subset of

C^{M}

. If

Σ \subseteq C (C)

with

Σ^{M} \subseteq W (C)

, then

Σ

is finite.

This statement can be quantified if

C

is presented as a curve in some projective space. Using BÃ©zout's Theorem, one can show that

# Σ

is bounded from above in terms of the degrees of

C

and

W

. In our application, both degrees will be uniformly bounded as all varieties arise in algebraic families. This ultimately leads to the desired uniformity estimates.

The second case is if the fiber of

D (C_{g}^{[M + 1]}) ∖ U \to A_{g}

above

τ (s)

has dimension at least

M + 1

. For dimension reasons,

s

lies in a proper subvariety

S

M_{g}

. Here we apply induction on the dimension and replace

M_{g}

by its subvariety

S

This completes the proof sketch.

KÃ¼hne [39] combined ideas from equidistribution with the approach laid out in [24] to get a suitable uniform estimate for

# B (P_{0}, r)

without the restriction (5.3) on

h_{M_{g}} (s)

. Yuan's Theorem 1.1 [66] does so as well, but he follows a different approach. He obtains a more general estimate that works also in the function field setting and allows for a larger

R

6. HYPERELLIPTIC CURVES

A hyperelliptic curve is a smooth curve of genus at least 2 that admits a degree 2 morphism to the projective line. Hyperelliptic curves have particularly simple planar models. Indeed, if the base field is a number field

F

, then a hyperelliptic curve of genus

g

can be
represented by a hyperelliptic equation

Y^{2} = f (X)

with

f \in F [X]

monic and square-free of degree

2 g + 1

2 g + 2

In this section we determine consequences of Theorem 1.12 for hyperelliptic curves. Our aim is to leave the world of curves and Jacobians and to present a bound for the number of rational solutions of

Y^{2} = f (X)

that can be expressed in terms of

f

. We refer to Section 6 of [23] for a similar example in a 1-parameter family of hyperelliptic curves.

To keep technicalities to a minimum, we assume that our base field is

F = Q

and that

f \in Z [X]

is monic of degree

d = 2 g + 1

and factors into linear factors in

Q [X]

. The curve represented by the hyperelliptic equation has a marked Weierstrass point "at infinity." These assumptions can be loosened with some extra effort. For example, if

f

does not factor in

Q [X]

, then the class number of the splitting field will play a part.

Say,

f = X^{d} + f_{d - 1} X^{d - 1} + \dots + f_{0}

. By the assumption above,

f = (X - α_{1}) \dots (X - α_{d})

with

α_{1}, \dots, α_{d} \in Q

which are necessarily integers. The discriminant of

f

Δ_{f} = \prod_{1 \leq i < j \leq d} {(α_{j} - α_{i})}^{2} \in Z ∖ {0}

The Mordell Conjecture applied to the hyperelliptic curve represented by

Y^{2} = f (X)

states

# {(x, y) \in Q^{2} : y^{2} = f (x)} < \infty

We have the following estimate for the cardinality. Below

ω (n)

denotes the number of distinct prime divisors of

n \in Z ∖ {0}

Theorem 6.1. Let

g \geq 2

. There exist

c (g) > 1

and

c^{'} (g) > 0

with the following property. Suppose

f \in Z [X]

is monic of degree

2 g + 1

, square-free, and factors into linear factors in

Q [X]

. Then

\begin{matrix} (6.1) & # {(x, y) \in Q^{2} : y^{2} = f (x)} \leq c^{'} (g) c (g)^{ω (Δ_{f})} \end{matrix}

Proof. The hyperelliptic equation represents a curve

C

defined over

Q

of genus

g

p

is a prime number with

p ∤ Δ_{f}

, then the

α_{i}

are pairwise distinct modulo

p

. If

p

is also odd, then the equation

Y^{2} = f (X)

reduced modulo

p

defines a hyperelliptic curve over

F_{p}

. So

C

has good reduction at all primes that do not divide

2 Δ_{f}

. Thus the Jacobian

Jac (C)

has good reduction at the same primes.

We may embed

C

into its Jacobian

Jac (C)

by sending the marked Weierstrass point to 0 . Each root

α_{i}

f

corresponds to a rational point in

C (Q)

and it is sent to a point of order 2 in

Jac (C)

. Moreover, these points generate the 2-torsion in

Jac (C)_{tors}

. In particular, all points of order 2 in

Jac (C)_{tors}

are rational.

Next we bound the rank of

Jac (C) (Q)

from above. Indeed, we could use the work of Ooe-Top [49] or [37, THEOREM c.1.9]. The latter applied to

Jac (C), k = Q, m = 2

, and

S

the prime divisors of

2 Δ_{f}

yields

rk Jac (C) (Q) \leq 2 g # S \leq 2 g ω (2 Δ_{f}) \leq 2 g + 2 g ω (Δ_{f})

Here we use that

Q

has a trivial class group; in a more general setup, the class group of the
splitting field of

f

will enter at this point. The estimate (6.1) follows from Theorem 1.12 in the case

F = Q

with adjusted constants.

It is tempting to average (6.1) over the

f

bounded in a suitable way, e.g., by bounding the maximal modulus of the roots by a parameter

X

. As pointed out to the author by Christian Elsholtz and Martin Widmer, this average will be unbounded as

X \to \infty

ACKNOWLEDGMENTS

Much of the work presented here was obtained together with my coauthors. In particular, I thank Serge Cantat, Vesselin Dimitrov, Ziyang Gao, and Junyi Xie for the fruitful collaboration. I also thank Gabriel Dill, Ziyang Gao, Lars KÃ¼hne, and David Masser for comments on this survey.

FUNDING

The author has received funding from the Swiss National Science Foundation project nà¹ 200020_184623.

REFERENCES

[1] D. Abramovich, UniformitÃ© des points rationnels des courbes algÃ©briques sur les extensions quadratiques et cubiques. C. R. Acad. Sci. Paris SÃ©r. I Math. 321 (1995), no. 6, 755-758.

[2] L. Alpoge, The average number of rational points on genus two curves is bounded. 2018, arXiv:1804.05859.

[3] Y. AndrÃ©, P. Corvaja, and U. Zannier, The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao). Invent. Math. 222 (2020), 161-202.

[4] B. Bakker and J. Tsimerman, Lectures on the Ax-Schanuel conjecture. In Arithmetic geometry of logarithmic pairs and hyperbolicity of moduli spaces, edited by M. H. Nicole, pp. 1-68, CRM Short Courses, Springer, 2020.

[5] J. Balakrishnan, N. Dogra, J. S. MÃ¼ller, J. Tuitman, and J. Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13. Ann. of Math. (2) 189 (2019), no. 3, 885-944.

[6] J. S. Balakrishnan, A. J. Best, F. Bianchi, B. Lawrence, J. S. MÃ¼ller, N. Triantafillou, and J. Vonk, Two recent

p

-adic approaches towards the (effective) Mordell Conjecture. In Arithmetic L-functions and differential geometric methods, edited by P. Charollois, G. Freixas i Montplet, V. Maillot, pp. 31-74, Progr. Math. 338, BirkhÃ¤user, 2021.

[7] D. Bertrand, Manin's theorem of the kernel: a remark on a paper of C-L. Chai, preprint, 2008, https://webusers.imj-prg.fr/ daniel.bertrand/Recherche/rpdf/ Manin_Chai.pdf.

[8] E. Bombieri, The Mordell Conjecture revisited. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615-640.

[9] E. Bombieri and W. Gubler, Heights in diophantine geometry. Cambridge University Press, 2006.

[10] S. Cantat, Z. Gao, P. Habegger, and J. Xie, The geometric Bogomolov conjecture. Duke Math. J. 170 (2021), no. 2, 247-277.

[11] L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points. J. Amer. Math. Soc. 10 (1997), no. 1, 1-35.

[12] L. Caporaso, J. Harris, and B. Mazur, Corrections to uniformity of rational points. 2021, arXiv:2012.14461.

[13] C. Chabauty, Sur les points rationnels des courbes algÃ©briques de genre supÃ©rieur Ã l'unitÃ©. C. R. Acad. Sci. Paris 212 (1941), 882-885.

[14] S. Checcoli, F. Veneziano, and E. Viada, The explicit Mordell Conjecture for families of curves. Forum Math. Sigma 7 (2019), e31, 62 pp.

[15] R. F. Coleman, Effective Chabauty. Duke Math. J. 52 (1985), no. 3, 765-770.

[16] S. David, M. Nakamaye, and P. Philippon, Bornes uniformes pour le nombre de points rationnels de certaines courbes. In Diophantine geometry, pp. 143-164, CRM Series 4, Ed. Norm, Pisa, 2007.

[17] S. David and P. Philippon, Minorations des hauteurs normalisÃ©es des sous-variÃ©tÃ©s de variÃ©tÃ©s abeliennes. II. Comment. Math. Helv. 77 (2002), no. 4, 639-700.

[18] S. David and P. Philippon, Minorations des hauteurs normalisÃ©es des sous-variÃ©tÃ©s des puissances des courbes elliptiques. Int. Math. Res. Pap. 3 (2007), rpm006,

113 p p

[19] T. de Diego, Points rationnels sur les familles de courbes de genre au moins 2. J. Number Theory 67 (1997), no. 1, 85-114.

[20] L. DeMarco, H. Krieger, and H. Ye, Uniform Manin-Mumford for a family of genus 2 curves. Ann. of Math. 191 (2020), 949-1001.

[21] L. DeMarco and N. M. Mavraki, Variation of canonical height and equidistribution. Amer. J. Math. 142 (2020), no. 2, 443-473.

[22] V. Dimitrov, Z. Gao, and P. Habegger, A consequence of the relative Bogomolov conjecture. J. Number Theory, 230 (2022), 146-160.

[23] V. Dimitrov, Z. Gao, and P. Habegger, Uniform bound for the number of rational points on a pencil of curves. Int. Math. Res. Not. IMRN 2 (2021), 1138-1159.

[24] V. Dimitrov, Z. Gao, and P. Habegger, Uniformity in Mordell-Lang for curves. Ann. of Math. (2) 194 (2021), no. 1, 237-298.

[25] G. Faltings, EndlichkeitssÃ¤tze fÃ¼r abelsche VarietÃ¤ten Ã¼ber ZahlkÃ¶rpern. Invent. Math. 73 (1983), 349-366.

[26] G. Faltings, Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991), no. 3, 549-576.

[27] G. Faltings, The general case of S. Lang's conjecture. In Barsotti symposium in algebraic geometry (Abano Terme, 1991), pp. 175-182, Perspect. Math. 15, Academic Press, San Diego, CA, 1994.

[28] Z. Gao, A special point problem of AndrÃ©-Pink-Zannier in the universal family of Abelian varieties. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 1, 231-266.

[29] Z. Gao, Generic rank of Betti map and unlikely intersections. Compos. Math. 156 (2020), no. 12, 2469-2509.

[30] Z. Gao, Mixed Ax-Schanuel for the universal abelian varieties and some applications. Compos. Math. 156 (2020), no. 11, 2263-2297.

[31] Z. Gao, Recent developments of the uniform Mordell-Lang conjecture. 2021, arXiv:2104.03431.

[32] Z. Gao, T. Ge, and L. KÃ¼hne, The uniform Mordell-Lang conjecture. 2021, arXiv:2105.15085.

[33] Z. Gao and P. Habegger, Heights in families of Abelian varieties and the geometric Bogomolov conjecture. Ann. of Math. 189 (2019), no. 2, 527-604.

[34] P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. Invent. Math. 176 (2009), no. 2, 405-447.

[35] P. Habegger, Special points on fibered powers of elliptic surfaces. J. Reine Angew. Math. 685 (2013), 143-179.

[36] M. Hindry, Autour d'une conjecture de Serge Lang. Invent. Math. 94 (1988), no. 3, 575-603.

[37] M. Hindry and J. H. Silverman, Diophantine geometry: an introduction. Springer, 2000 .

[38] E. Katz, J. Rabinoff, and D. Zureick-Brown, Uniform bounds for the number of rational points on curves of small Mordell-Weil rank. Duke Math. J. 165 (2016), no. 16, 3189-3240.

[39] L. KÃ¼hne, Equidistribution in families of Abelian varieties and uniformity. 2021, arXiv:2101.10272.

[40] L. KÃ¼hne, The relative Bogomolov conjecture for fibered products of elliptic surfaces. 2021, arXiv:2103.06203.

[41] B. Lawrence and A. Venkatesh, Diophantine problems and

p

-adic period mappings. Invent. Math. 221 (2020), no. 3, 893-999.

[42] R. Lazarsfeld, Positivity in algebraic geometry. I. Ergeb. Math. Grenzgeb. (3). Ser. Mod. Surveys Math. [Res. Math. Relat. Areas (3)] 48, Springer, Berlin, 2004.

[43] D. Masser, Specializations of finitely generated subgroups of abelian varieties. Trans. Amer. Math. Soc. 311 (1989), no. 1, 413-424.

[44] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves. Math. Ann. 352 (2012), no. 2, 453-484.

[45] B. Mazur, Arithmetic on curves. Bull. Amer. Math. Soc. 14 (1986), no. 2, 207-259.

[46] B. Mazur, Abelian varieties and the Mordell-Lang conjecture. In Model theory, algebra, and geometry, pp. 199-227, Math. Sci. Res. Inst. Publ. 39, Cambridge University Press, Cambridge, 2000.

[47] N. Mok, J. Pila, and J. Tsimerman, Ax-Schanuel for Shimura varieties. Ann. of Math. (2) 189 (2019), no. 3, 945-978.

[48] L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Camb. Philos. Soc. 21 (1922), 179-192.

[49] T. Ooe and J. Top, On the Mordell-Weil rank of an abelian variety over a number field. J. Pure Appl. Algebra 58 (1989), no. 3, 261-265.

[50] P. L. Pacelli, Uniform boundedness for rational points. Duke Math. J. 88 (1997), no.

1, 77 - 102

[51] J. Pila and A. J. Wilkie, The rational points of a definable set. Duke Math. J. 133 (2006), no. 3, 591-616.

[52] M. Raynaud, Courbes sur une variÃ©tÃ© abÃ©lienne et points de torsion. Invent. Math. 71 (1983), no. 1, 207-233.

[53] M. Raynaud, Sous-variÃ©tÃ©s d'une variÃ©tÃ© abÃ©lienne et points de torsion. In Arithmetic and geometry, Vol. I, pp. 327-352, Progr. Math. 35, BirkhÃ¤user Boston, Boston, MA, 1983.

[54] G. RÃ©mond, DÃ©compte dans une conjecture de Lang. Invent. Math. 142 (2000), no. 3, 513-545.

[55] G. RÃ©mond, InÃ©galitÃ© de Vojta en dimension supÃ©rieure. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 29 (2000), no. 1, 101-151.

[56] G. RÃ©mond, Borne gÃ©nÃ©rique pour le problÃ¨me de Mordell-Lang. Manuscripta Math. 118 (2005), no. 1, 85-97.

[57] J. H. Silverman, Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math. 342 (1983), 197-211.

[58] M. Stoll, Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 3, 923-956.

[59] L. Szpiro, Un peu d'effectivitÃ©. AstÃ©risque 127 (1985), 275-287.

[60] L. Szpiro, E. Ullmo, and S. Zhang, Ã‰quirÃ©partition des petits points. Invent. Math. 127 (1997), no. 2, 337-347.

[61] E. Ullmo, PositivitÃ© et discrÃ¨tion des points algÃ©briques des courbes. Ann. of Math. (2) 147 (1998), no. 1, 167-179.

[62] P. Vojta, Siegel's theorem in the compact case. Ann. of Math. (2) 133 (1991), no. 3, 509-548.

[63] Y. Xie and X. Yuan, Geometric Bogomolov conjecture in arbitrary characteristics. 2021, arXiv:2108.09722.

[64] K. Yamaki, Survey on the geometric Bogomolov conjecture. In Actes de la ConfÃ©rence "Non-Archimedean analytic geometry: theory and practice", pp. 137-193, Publ. Math. BesanÃ§on AlgÃ¨bre ThÃ©orie Nr. 2017/1, Presses Univ. Franche-ComtÃ©, BesanÃ§on, 2017.

[65] X. Yuan, Big line bundles over arithmetic varieties. Invent. Math. 173 (2008), no. 3, 603-649.

[66] X. Yuan, Arithmetic bigness and a uniform Bogomolov-type result. 2021, arXiv:2108.05625.

[67] X. Yuan and S. Zhang, Adelic line bundles over quasi-projective varieties. 2021, arXiv:2105.13587.

[68] U. Zannier, Some problems of unlikely intersections in arithmetic and geometry. Ann. of Math. Stud. 181, Princeton University Press, Princeton, NJ, 2012.

[69] J. G. Zarhin and J. I. Manin, Height on families of abelian varieties. Mat. Sb. (N.S.) 89 (1972), no. 131, 171-181, 349 .

[70] S. Zhang, Small points and adelic metrics. J. Algebraic Geom. 4 (1995), no. 2, 281-300.

[71] S. Zhang, Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, 159-165.

[72] S. Zhang, Small points and Arakelov theory. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., pp. 217-225, 1998. Extra Vol. II.

[73] H. Zimmer, On the difference of the Weil height and the NÃ©ron-Tate height. Math. Z. 147 (1976), no. 1, 35-51.

PHILIPP HABEGGER

Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland, philipp.habegger@unibas.ch

THETA LIFTING AND LANGLANDS FUNCTORIALITY

ATSUSHI ICHINO

ABSTRACT

We review various aspects of theta lifting and its role in studying Langlands functoriality. In particular, we discuss realizations of the Jacquet-Langlands correspondence and the Shimura-Waldspurger correspondence in terms of theta lifting and their arithmetic applications.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11F70; Secondary 11F27, 11F67, 22E50

KEYWORDS

Automorphic representations, Langlands functoriality, theta lifting

1. INTRODUCTION

Langlands functoriality is a principle relating two different kinds of automorphic forms and plays a pivotal role in number theory. Before Langlands formulated this principle in [42], this phenomenon was already observed in the following classical example discovered by Eichler [15] and developed by Shimizu [48]. Consider the space

S_{k} (Γ_{0} (N))

of elliptic cusp forms of weight

k

and level

N

, where

k

and

N

are positive integers and

Γ_{0} (N)

is the congruence subgroup given by

Γ_{0} (N) = {(\begin{array}{ll} a & b \\ c & d \end{array}) \in {S L}_{2} (Z) | c \equiv 0 mod N}

This space consists of holomorphic functions

f

on the upper half-plane

S

which satisfy

f (γ z) = j (γ, z)^{k} f (z)

for all

γ \in Γ_{0} (N)

and

z \in S

and which vanish at all cusps. Here

{S L}_{2} (R)

acts on

S

by linear fractional transformations and

j ((\begin{array}{ll} a & b \\ c & d \end{array}), z) = c z + d

is the factor of automorphy. It is also equipped with the action of Hecke operators

T_{n}

for all positive integers

n

, which is a central tool in the arithmetic study of automorphic forms. On the other hand, to every indefinite quaternion division algebra

B

over

Q

, we may associate a space

S_{k} (Γ_{B})

of modular forms, where

Γ_{B}

is the group of norm-one elements in

B

. Namely, this space is defined similarly by replacing

Γ_{0} (N)

Γ_{B}

(which can be regarded as a subgroup of

{(B \otimes_{Q} R)}^{\times} ≅ {G L}_{2} (R))

and is equipped with the action of Hecke operators

T_{n}^{B}

. Now assume that

N

is the product of an even number of distinct primes and

B

is ramified precisely at the primes dividing

N

. Then by the works of Eichler and Shimizu, the trace of

T_{n}^{B}

S_{k} (Γ_{B})

coincides with the trace of

T_{n}

on the new part of

S_{k} (Γ_{0} (N))

for all

n

prime to

N

This remarkable relation was thoroughly studied by Jacquet-Langlands [37] in the framework of automorphic representations. Let

F

be a number field with adÃ¨le ring

A

. Let

B

be a quaternion division algebra over

F

. Then Jacquet-Langlands proved that for any irreducible automorphic representation

π^{B} ≅ \otimes_{v} π_{v}^{B}

B^{\times} (A)

, there exists a unique irreducible automorphic representation

π ≅ \otimes_{v} π_{v}

{G L}_{2} (A)

such that

π_{v} ≅ π_{v}^{B}

for almost all places

v

F

. Moreover, they described the image of this map

π^{B} \mapsto π

precisely.

The Jacquet-Langlands correspondence gives a basic example of Langlands functoriality. To explain this, let

G

be a connected reductive group over

F

. Let

^{L} G

be the

L

-group of

G

, which was introduced by Langlands and which should govern automorphic representations of

G (A)

. Explicitly,

^{L} G

is defined as a semiproduct

\hat{G} ⋊ Γ_{F}

, where

\hat{G}

is the complex dual group of

G, Γ_{F} = Gal (\bar{F} / F)

is the absolute Galois group of

F

, and the action of

Γ_{F}

\hat{G}

is inherited from the action of

Γ_{F}

on the root datum of

G

. To motivate it, let us admit for the moment the existence of the hypothetical Langlands group

L_{F}

over

F

, which is equipped with a surjection

L_{F} \to Γ_{F}

. Then it is conjectured that irreducible automorphic representations of

G (A)

are classified in terms of certain

L

-homomorphisms

L_{F} \to^{L} G

, i.e., homomorphisms commuting with the projections to

Γ_{F}

. (Strictly speaking, we consider here packets of tempered automorphic representations.) Now suppose that we have another connected reductive quasisplit group

G^{'}

over

F

and an

L

-homomorphism

r :^{L} G \to^{L} G^{'}

Let

π

be an irreducible automorphic representation of

G (A)

which should correspond to an

L

-homomorphism

ϕ : L_{F} \to^{L} G

Then Langlands functoriality predicts the existence of an irreducible automorphic representation

π^{'}

G^{'} (A)

which should correspond to the

L

-homomorphism

r \circ ϕ : L_{F} \to^{L} G^{'}

This conjectural relation between

π

and

π^{'}

can be formulated without assuming the existence of

L_{F}

as follows. Recall that for almost all places

v

F

, the local component

π_{v}

π

v

is unramified, so that it determines and is determined by a

\hat{G}

-conjugacy class

c (π_{v})

^{L} G

via the Satake isomorphism. Then

π^{'}

should satisfy

c (π_{v}^{'}) = r (c (π_{v}))

for almost all

v

. Note that the Jacquet-Langlands correspondence mentioned above is the special case when

G = B^{\times}, G^{'} = {G L}_{2}

(so that

^{L} G =^{L} G^{'} = {G L}_{2} (C) \times Γ_{F}

), and

r

is the identity map.

Although Langlands functoriality is out of reach in general, it led to substantial developments in the theory of automorphic forms. For example, the trace formula was developed by Arthur to study automorphic representations, culminating in his book [1] which establishes the case when

G

is a symplectic group or a quasisplit special orthogonal group,

G^{'}

is a general linear group, and

r

is the standard embedding. There are also other methods to attack Langlands functoriality, such as the converse theorem [12,13], the automorphic descent [24], and the theta lifting. In this report, we will discuss various aspects of the theta lifting, which can be viewed as an explicit realization in the case when

(G, G^{'})

is a certain pair of classical groups.

2. THETA LIFTING

In this section, we recall the notion of the theta lifting, with emphasis on the realization of the Jacquet-Langlands correspondence. We also review some of its applications to explicit formulas for automorphic periods in terms of special values of

L

-functions.

2.1. Basic definitions and properties

Let

F

be a number field with adÃ¨le ring

A = A_{F}

. Let

W

be a symplectic space over

F

equipped with a nondegenerate bilinear alternating form

(\cdot, \cdot)_{W}

and let

Sp (W)

denote the symplectic group of

W

. Similarly, let

V

be a quadratic space over

F

equipped with a nondegenerate bilinear symmetric form

(\cdot, \cdot)_{V}

and let

O (V)

denote the orthogonal group of

V

. Then the pair

(Sp (W), O (V))

is an example of a reductive dual pair introduced by Howe [30]. Namely, if we consider the symplectic space

W = W \otimes_{F} V

equipped with the form

(\cdot, \cdot)_{W} \otimes (\cdot, \cdot)_{V}

and the natural homomorphism

Sp (W) \times O (V) \to S p (W)

then

S p (W)

and

O (V)

are mutual commutants in

S p (W)

Roughly speaking, the theta lifting is an integral transform with kernel given by a particular automorphic form on

Sp (W) (A)

restricted to

Sp (W) (A) \times O (V) (A)

. To be precise, we need to consider the metaplectic group

M p (W) (A)

, which is a nontrivial topological central extension

1 \to {\pm 1} \to Mp (W) (A) \to Sp (W) (A) \to 1

(Here we have abused notation since

Mp (W) (A)

is not the group of

A

-valued points of an algebraic group over

F

.) This extension splits over

Sp (W) (F)

canonically, so that we may speak of automorphic forms on

Mp (W) (A)

. We are interested in a particular representation

ω

Mp (W) (A)

(depending on a choice of a nontrivial additive character of

A / F

), called the Weil representation [61], which is a representation theoretic incarnation of theta functions. This representation has an automorphic realization, i.e., there is an

Mp (W) (A)

-equivariant

map φ \mapsto θ_{φ}

from

ω

to the space of automorphic forms on

Mp (W) (A)

. On the other hand, there exists a dotted arrow making the following diagram commute:

(Note that it descends to a homomorphism from the bottom left corner if and only if

\dim V

is even.) Thus we may regard

θ_{φ}

as an automorphic form on

Mp (W) (A) \times O (V) (A)

by restriction and associate to an automorphic form

f

Mp (W) (A)

an automorphic form

θ_{φ} (f)

O (V) (A)

by setting

θ_{φ} (f) (h) = \int_{Sp (W) (F) ∖ Mp (W) (A)} θ_{φ} (g, h) \bar{f (g)} d g

provided the integral converges, e.g., if

f

is cuspidal.

For any irreducible cuspidal automorphic representation

π

Mp (W) (A)

, we define the theta lift

θ (π)

π

as the automorphic representation of

O (V) (A)

spanned by

θ_{φ} (f)

for all

φ \in ω

and

f \in π

. We only consider the case when

π

descends (resp. does not descend)
to a representation of

Sp (W) (A)

\dim V

is even (resp. odd); otherwise

θ (π)

is always zero. To describe

θ (π)

, we need to introduce the local analog of the theta lifting. First, note that the map

(φ, f) \mapsto θ_{φ} (f)

defines an element in

{Hom}_{Mp (W) (A) \times O (V) (A)} (ω \otimes \bar{π}, θ (π)) ≅ {Hom}_{Mp (W) (A) \times O (V) (A)} (ω, π \otimes θ (π))

since

π

is unitary. Recall that

ω

can be regarded as the restricted tensor product of the local Weil representations

ω_{v}

Mp (W) (F_{v})

via the surjection

\prod_{v}^{'} Mp (W) (F_{v}) \to Mp (W) (A)

, where

Mp (W) (F_{v})

is the metaplectic cover of

Sp (W) (F_{v})

. Similarly,

π

can be decomposed as

π ≅ \otimes_{v} π_{v}

, where

π_{v}

is an irreducible representation of

Mp (W) (F_{v})

. We define the local theta lift

θ (π_{v})

π_{v}

as an irreducible representation of

O (V) (F_{v})

such that

{Hom}_{Mp (W) (F_{v}) \times O (V) (F_{v})} (ω_{v}, π_{v} \otimes θ (π_{v})) \neq 0

which is unique (if it exists) by the Howe duality

[23, 31, 55]

. (When such a representation does not exist, we interpret

θ (π_{v})

as zero.) Now assume that

θ (π)

is nonzero and cuspidal. Then it follows from the Howe duality that

θ (π)

is irreducible and can be decomposed as

θ (π) ≅ \otimes_{v} θ (π_{v})

Remark 2.1. We may extend the Weil representation and define the theta lifting for the pair

(GSp (W), GO (V))

, where

GSp (W)

and

GO (V)

are the similitude groups of

W

and

V

, respectively.

2.2. Explicit realization of the Jacquet-Langlands correspondence

From now on, we mainly consider the case when

\dim W = 2, \dim V = 4

and the discriminant of

V

is trivial. Then we may identify

W

with the space

F^{2}

, equipped with the form

{((x_{1}, x_{2}), (y_{1}, y_{2}))}_{W} = x_{1} y_{2} - x_{2} y_{1}

, so that

GSp (W) = {G L}_{2}

We may also identify

V

with a quaternion algebra

B

over

F

equipped with the form

(x, y)_{V} = {Tr}_{B / F} (x y^{*})

, where

{Tr}_{B / F}

is the reduced trace and

*

is the main involution, so that

GO (V)^{0} = (B^{\times} \times B^{\times}) / F^{\times}

Here

GO (V)^{0}

is the identity component of

GO (V)

and

B^{\times} \times B^{\times}

acts on

V

by left and right multiplication.

Let

π

be an irreducible cuspidal automorphic representation of

{G L}_{2} (A)

. We regard the theta lift

θ (π)

π

(restricted to

GO (V)^{0} (A)

) as an automorphic representation of

B^{\times} (A) \times B^{\times} (A)

. Then Shimizu [49] proved that

θ (π) = π^{B} \otimes π^{B}

where

π^{B}

is the Jacquet-Langlands transfer of

π

B^{\times} (A)

. (When

π

does not transfer to

B^{\times} (A)

, we interpret

π^{B}

as zero.)

Remark 2.2. In [36], we gave the following variant of the above realization. Let

B, B_{1}, B_{2}

be three quaternion division algebras over

F

such that

B = B_{1} \cdot B_{2}

in the Brauer group. We consider a 1-dimensional Hermitian space

W

over

B

and a 2-dimensional skew-Hermitian space

V

over

B

such that

G U (W) = B^{\times}, G U (V)^{0} = (B_{1}^{\times} \times B_{2}^{\times}) / F^{\times}

where

G U (W)

and

G U (V)

are the unitary similitude groups of

W

and

V

, respectively. Let

π^{B}

be an irreducible automorphic representation of

B^{\times} (A)

such that its Jacquet-Langlands transfer to

{G L}_{2} (A)

is cuspidal. Then we have

θ (π^{B}) = π^{B_{1}} \otimes π^{B_{2}}

where

π^{B_{1}}

and

π^{B_{2}}

are the Jacquet-Langlands transfers of

π^{B}

B_{1}^{\times} (A)

and

B_{2}^{\times} (A)

, respectively. We believe that this realization is useful to study integral period relations.

2.3. Seesaw identities

One of the advantages of the theta lifting is that it produces various period relations in a simple way, which was observed by Kudla [39]. Suppose that we have two reductive dual pairs

(G, H)

and

(G^{'}, H^{'})

in the same symplectic group such that

G \subset G^{'}

and

H \supset H^{'}

. This can be illustrated by the following picture, called a seesaw diagram:

Let

f

and

f^{'}

be automorphic forms on

G (A)

and

H^{'} (A)

, respectively. Then the theta lifting produces automorphic forms

θ_{φ} (f)

and

θ_{φ} (f^{'})

H (A)

and

G^{'} (A)

, respectively, and the so-called seesaw identity

\begin{aligned} ⟨ {θ_{φ} (f) |}_{H^{'} (A)}, f^{'} ⟩ & = \int_{H^{'} (F) ∖ H^{'} (A)} θ_{φ} (f) (h^{'}) \bar{f^{'} (h^{'})} d h^{'} \\ = \int_{G (F) ∖ G (A)} \int_{H^{'} (F) ∖ H^{'} (A)} θ_{φ} (g, h^{'}) \bar{f (g) f^{'} (h^{'})} d g d h^{'} \\ = \int_{G (F) ∖ G (A)} \bar{f (g)} θ_{φ} (f^{'}) (g) d g = \bar{⟨ f, {θ_{φ} (f^{'}) |}_{G (A)})} \end{aligned}

provided the double integral converges absolutely. Here

⟨ \cdot, \cdot ⟩

denotes the Petersson inner product.

As an example of this identity, we recall Waldspurger's formula for torus periods. We keep the setup of the previous subsection. Fix a quadratic extension

E

F

which embeds into

B

and write

B = E \oplus E j

with a trace zero element

j

B

. Let

V = V_{1} \oplus V_{2}

be the corresponding decomposition of quadratic spaces, so that

GO {(V_{1})}^{0} = G O {(V_{2})}^{0} = E^{\times}

Then the identification

W \otimes_{F} V = (W \otimes_{F} V_{1}) \oplus (W \otimes_{F} V_{2})

gives rise to the following seesaw diagram:

Let

π^{B}

be an irreducible automorphic representation of

B^{\times} (A)

such that its JacquetLanglands transfer

π

{G L}_{2} (A)

is cuspidal. Let

χ

be an automorphic character of

A_{E}^{\times}

. Assume that the product of the central character of

π^{B}

and the restriction of

χ

A_{F}^{\times}

is trivial and consider the torus period

P (f, χ) = \int_{E^{\times} A_{F}^{\times} ∖ A_{E}^{\times}} f (h) χ (h) d h

for a decomposable vector

f \in π^{B}

. Then using the above seesaw diagram, Waldspurger [54] proved that

| P (f, χ) |^{2} = \frac{1}{4} \frac{ζ (2) L (1 / 2, π_{E} \times χ)}{L (1, π, Ad) L (1, μ_{E / F})} \prod_{v} α_{v} (f_{v}, χ_{v})

where

$ζ (s)$ is the completed Dedekind zeta function of $F$ ,
$L (s, π_{E} \times χ)$ is the standard $L$ -function of the base change $π_{E}$ of $π$ to ${G L}_{2} (A_{E})$ twisted by $χ$ ,
$L (s, π$ , Ad $)$ is the adjoint $L$ -function of $π$ ,
$L (s, μ_{E / F})$ is the Hecke $L$ -function of the quadratic automorphic character $μ_{E / F}$ of $A_{F}^{\times}$ associated to $E / F$ by class field theory,
$α_{v} (f_{v}, χ_{v})$ is a certain normalized local integral of matrix coefficients.

As another example, we consider the 6-dimensional symplectic space

W^{'} = W^{3}

over

F

. Then the identification

W^{'} \otimes_{F} V = {(W \otimes_{F} V)}^{3}

gives rise to the following seesaw diagram:

Let

π_{1}^{B}, π_{2}^{B}, π_{3}^{B}

be irreducible automorphic representations of

B^{\times} (A)

such that their Jacquet-Langlands transfers

π_{1}, π_{2}, π_{3}

{G L}_{2} (A)

are cuspidal. Assume that the product of the central characters of

π_{1}^{B}, π_{2}^{B}, π_{3}^{B}

is trivial and consider the trilinear period

P (f_{1}, f_{2}, f_{3}) = \int_{B^{\times} A^{\times} ∖ B^{\times} (A)} f_{1} (h) f_{2} (h) f_{3} (h) d h

for decomposable vectors

f_{1} \in π_{1}^{B}, f_{2} \in π_{2}^{B}, f_{3} \in π_{3}^{B}

. Then following the work of HarrisKudla [27] and using the above seesaw diagram, we proved in [32] that

{| P (f_{1}, f_{2}, f_{3}) |}^{2} = \frac{1}{8} \frac{ζ (2)^{2} L (1 / 2, π_{1} \times π_{2} \times π_{3})}{L (1, π_{1}, Ad) L (1, π_{2}, Ad) L (1, π_{3}, Ad)} \prod_{v} α_{v} (f_{1, v}, f_{2, v}, f_{3, v})

where

$L (s, π_{1} \times π_{2} \times π_{3})$ is the triple product $L$ -function of $π_{1}, π_{2}, π_{3}$ ,
$α_{v} (f_{1, v}, f_{2, v}, f_{3, v})$ is a certain normalized local integral of matrix coefficients.

Remark 2.3. The above two formulas are special cases of the Gross-Prasad conjecture [25,26] and its refinement [33]. This conjecture (for special orthogonal groups) was extended to all classical groups by Gan-Gross-Prasad [17], and after the breakthrough of Zhang [63,64], the global conjecture for unitary groups has been proved in a series of works [8-11,62] using the relative trace formula. We should also mention the stunning work of Waldspurger [57-60], which led to the proof of the local Gan-Gross-Prasad conjecture for Bessel models [5-7,45] and Fourier-Jacobi models [2,19] in the

p

-adic case, where the theta lifting is used to deduce the latter from the former.

3. THE SHIMURA-WALDSPURGER CORRESPONDENCE

In this section, we review some applications of the theta lifting to automorphic forms on metaplectic groups.

3.1. Modular forms of half-integral weight

The theta function

θ (z) = \sum_{n = - \infty}^{\infty} e^{2 π i n^{2} z}

is a modular form of weight

1 / 2

and its significance is well known. Thus it is natural to study modular forms of half-integral weight, but Hecke [28, P. 152] realized the difficulty in developing the arithmetic theory; the Hecke operator

T_{n}

is zero unless

n

is a square. In 1973, Shimura [50] revolutionized the theory of modular forms of half-integral weight by relating them to modular forms of integral weight, i.e., he constructed a modular form of weight

2 k

from a cusp form of weight

k + 1 / 2

by using the converse theorem, where

k

is a positive integer. Soon after the discovery of this correspondence, Niwa [46] and Shintani [51] gave an alternative construction using the theta lifting. This was further investigated by Waldspurger [53,56] in the framework of automorphic representations. Namely, he established a correspondence between automorphic representations of

{M p}_{2} (A)

(where

{M p}_{2} (A)

is the metaplectic cover of

{S L}_{2} (A))

and those of

{P G L}_{2} (A)

, which can be viewed as an example of Langlands functoriality.

3.2. Global correspondence

Now we discuss a generalization of the Shimura-Waldspurger correspondence to metaplectic groups of higher rank. Let

F

be a number field with adÃ¨le ring

A

. We denote by

{S p}_{2 n}

the symplectic group of rank

n

over

F

and by

{Mp}_{2 n} (A)

the metaplectic cover of

{S p}_{2 n} (A)

. Recall that this cover splits over

{S p}_{2 n} (F)

canonically, so that we may speak of the unitary representation of

{Mp}_{2 n} (A)

on the Hilbert space

L^{2} ({Sp}_{2 n} (F) ∖ {Mp}_{2 n} (A))

given by right translation. Since we are interested in genuine automorphic representations of

{Mp}_{2 n} (A)

, i.e., those which do not descend to representations of

{S p}_{2 n} (A)

, we only consider its subspace

L^{2} ({M p}_{2 n})

on which the central subgroup

{\pm 1}

acts by the nontrivial character. Write

L^{2} ({M p}_{2 n}) = L_{disc}^{2} ({M p}_{2 n}) \oplus L_{cont}^{2} ({M p}_{2 n})

for the decomposition into the discrete part and the continuous part. Then the theory of Eisenstein series gives an explicit description of

L_{cont}^{2} ({M p}_{2 n})

in terms of automorphic discrete spectra of proper Levi subgroups of

{M p}_{2 n}

, i.e.,

{G L}_{n_{1}} \times \dots \times {G L}_{n_{k}} \times {M p}_{2 n_{0}}

with

n_{1} + \dots + n_{k} + n_{0} = n

and

n_{0} < n

. Thus the problem is to describe the irreducible decomposition of

L_{disc}^{2} ({M p}_{2 n})

To attack this problem, it is better to divide it into two parts as follows:

(1) Describe the decomposition of

L_{disc}^{2} ({M p}_{2 n})

into near equivalence classes. Here we say that two irreducible genuine representations

π ≅ \otimes_{v} π_{v}

and

π^{'} ≅ \otimes_{v} π_{v}^{'}

{Mp}_{2 n} (A)

are nearly equivalent if

π_{v}

and

π_{v}^{'}

are equivalent for almost all places

v

F

. (In particular, if

π

and

π^{'}

are equivalent, then they are nearly equivalent.) Note that

π_{v}

is unramified for almost all

v

, so that it determines and is determined by a semisimple conjugacy class

c_{ψ_{v}} (π_{v})

{S p}_{2 n} (C)

(depending on a choice of a nontrivial additive character

ψ_{v}

F_{v}

) via the Satake isomorphism. In other words, the near equivalence classes of irreducible genuine representations of

{M p}_{2 n} (A)

can be parametrized by families of semisimple conjugacy classes

{c_{v}}_{v}

{S p}_{2 n} (C)

, where we identify two families if they are equal for almost all

v

. Thus we want to describe the families

{c_{v}}_{v}

which correspond to the near equivalence classes in

L_{disc}^{2} ({M p}_{2 n})

(2) Describe the irreducible decomposition of each near equivalence class. Namely, for any near equivalence class

C

L_{disc}^{2} ({M p}_{2 n})

and any irreducible genuine representation

π

{Mp}_{2 n} (A)

, we want to give an explicit formula for the multiplicity of

π

C

in terms of the classification of representations.

In [20], we solved (1) completely and (2) partially; we described the families

{c_{v}}_{v}

as above in terms of automorphic representations of general linear groups, and admitting that Arthur's endoscopic classification [1] can be extended to nonsplit odd special orthogonal groups, we established the multiplicity formula for the tempered part of

L_{disc}^{2} ({M p}_{2 n})

Now we state the first result precisely.

Theorem 3.1 ([20]). Fix a nontrivial additive character

ψ = \otimes_{v} ψ_{v}

A / F

. Then we have a decomposition

L_{disc}^{2} ({M p}_{2 n}) = ⨁_{ϕ} L_{ϕ}^{2} ({M p}_{2 n})

where

ϕ

runs over elliptic A-parameters for

{M p}_{2 n}

. Here an elliptic A-parameter for

{M p}_{2 n}

is defined to be a formal finite direct sum

⨁_{i} ϕ_{i} \otimes S_{d_{i}}

(which is a substitute for a hypothetical L-homomorphism

L_{F} \times {S L}_{2} (C) \to {S p}_{2 n} (C)

), where

$ϕ_{i}$ is an irreducible self-dual cuspidal automorphic representation of ${G L}_{n_{i}} (A)$ (which is hypothetically identified with an $n_{i}$ -dimensional irreducible representation of $L_{F}$ ),
$S_{d_{i}}$ is the $d_{i}$ -dimensional irreducible representation of ${S L}_{2} (C)$ ,
if $d_{i}$ is odd, then $ϕ_{i}$ is symplectic, i.e., the exterior square $L$ -function $L (s, ϕ_{i}, \land^{2})$ has a pole at $s = 1$ (and hence $n_{i}$ is even),
if $d_{i}$ is even, then $ϕ_{i}$ is orthogonal, i.e., the symmetric square L-function $L (s, ϕ_{i}$ , Sym $^{2})$ has a pole at $s = 1$ ,
if $i \neq j$ , then $(ϕ_{i}, d_{i}) \neq (ϕ_{j}, d_{j})$ ,
$\sum_{i} n_{i} d_{i} = 2 n$ .

Also,

L_{ϕ}^{2} ({M p}_{2 n})

is defined as the near equivalence class in

L_{disc}^{2} ({M p}_{2 n})

which corresponds to the family of semisimple conjugacy classes

{c_{v} (ϕ_{v})}_{v}

{S p}_{2 n} (C)

given as follows (so that any irreducible summand

π

L_{ϕ}^{2} ({M p}_{2 n})

satisfies

c_{ψ_{v}} (π_{v}) = c_{v} (ϕ_{v})

for almost all

v

). Suppose that

v

is finite and

ϕ_{i, v}

is unramified for all

i

. Let

c_{v} (ϕ_{i, v})

be the semisimple conjugacy class in

{G L}_{n_{i}} (C)

which corresponds to

ϕ_{i, v}

and put

Q_{v} (d) = (\begin{array}{llll} q_{v}^{(d - 1) / 2} \\ q_{v}^{(d - 3) / 2} \\ ⋱ \\ q_{v}^{- (d - 1) / 2} \end{array})

for any positive integer

d

, where

q_{v}

is the cardinality of the residue field of

F_{v}

. We regard

c_{v} (ϕ_{i, v}) \otimes Q_{v} (d_{i})

as a semisimple conjugacy class in

{S p}_{n_{i} d_{i}} (C)

. Then we set

c_{v} (ϕ_{v}) = ⨁_{i} c_{v} (ϕ_{i, v}) \otimes Q_{v} (d_{i})

To state the second result precisely, we need to introduce more notation. For each place

v

F

, let

W_{F_{v}}

be the Weil group of

F_{v}

and put

L_{F_{v}} = {\begin{cases} W_{F_{v}} & if v is infinite \\ W_{F_{v}} \times {S L}_{2} (C) & if v is finite \end{cases}

Let

ϕ = ⨁_{i} ϕ_{i} \otimes S_{d_{i}}

be an elliptic

A

-parameter for

{M p}_{2 n}

. We regard its local component

ϕ_{v} = ⨁_{i} ϕ_{i, v} \otimes S_{d_{i}}

v

as a local

A

-parameter

ϕ_{v} : L_{F_{v}} \times {S L}_{2} (C) \to {S p}_{2 n} (C)

via the local Langlands correspondence. Note that

c_{v} (ϕ_{v}) = ϕ_{v} ({Fr}_{v}, (\begin{array}{ll} q_{v}^{1 / 2} \\ q_{v}^{- 1 / 2} \end{array}))

for almost all

v

, where

{Fr}_{v}

is a Frobenius element at

v

. We denote by

ℑ_{ϕ_{v}}

the component group of the centralizer of

ϕ_{v}

{S p}_{2 n} (C)

, which is an elementary abelian 2-group, and by

ς_{ϕ} = ⨁_{i} (Z / 2 Z) a_{i}

the global component group of

ϕ

, which is formally defined as an elementary abelian 2-group with a basis

{a_{i}}_{i}

indexed by

{ϕ_{i} \otimes S_{d_{i}}}_{i}

. Then we have a natural homomorphism

S_{ϕ} \to S_{ϕ_{v}}

for all

v

. We also consider the compact abelian group

S_{ϕ, A} = \prod_{v} ς_{ϕ_{v}}

and the diagonal map

Δ : S_{ϕ} \to S_{ϕ, A}

Theorem 3.2 ([20]). Assume that

ϕ

is tempered, i.e.,

d_{i} = 1

for all

i

. Then we have a decomposition

L_{ϕ}^{2} ({M p}_{2 n}) ≅ ⨁_{η} m_{η} π_{η}

where

η = \otimes_{v} η_{v}

runs over continuous characters of

S_{ϕ, A}

. Here

π_{η}

is defined as the restricted tensor product of representations

π_{η_{v}}

in the local L-packets

Π_{ϕ_{v}} ({Mp}_{2 n} (F_{v}))

associated to

ϕ_{v}

(depending on

ψ_{v}

), which consist of irreducible genuine representations of

{Mp}_{2 n} (F_{v})

indexed by characters of

S_{ϕ_{v}}

. Also, if we define a character

ε_{ϕ}

ℑ_{ϕ}

ε_{ϕ} (a_{i}) = ε (1 / 2, ϕ_{i})

where

ε (s, ϕ_{i})

is the standard

ε

-function of

ϕ_{i}

, then

m_{η}

is given by

m_{η} = {\begin{cases} 1 & if η \circ Δ = ε_{ϕ} \\ 0 & otherwise \end{cases}

Remark 3.3. In fact, we gave another proof of the result of Waldspurger for

{M p}_{2}

[53, 56], noting that an irreducible cuspidal automorphic representation of

{G L}_{2} (A)

is symplectic if and only if its central character is trivial.

Remark 3.4. If we denote by

{S O}_{2 n + 1}

the split odd special orthogonal group of rank

n

over

F

and by

L_{disc}^{2} ({S O}_{2 n + 1})

the discrete part of

L^{2} ({S O}_{2 n + 1} (F) ∖ {S O}_{2 n + 1} (A))

, then the decomposition of

L_{disc}^{2} ({M p}_{2 n})

is similar to that of

L_{disc}^{2} ({S O}_{2 n + 1})

given by Arthur [1], except that the condition

η \circ Δ = ε_{ϕ}

in the former has to be replaced by

η \circ Δ = 1

in the latter.

In the proof of his result for

n = 1

, Waldspurger used the theta lifting between

{M p}_{2}

and (inner forms of)

{P G L}_{2} ≅ {S O}_{3}

. Thus, in general, it would be natural to use the theta lifting between

{M p}_{2 n}

and (inner forms of)

{S O}_{2 n + 1}

, and then transfer Arthur's endoscopic classification from

{S O}_{2 n + 1}

{M p}_{2 n}

. However, there is a serious obstacle in this approach. Indeed, if

π

is an irreducible genuine cuspidal automorphic representation of

{M p}_{2 n} (A)

and its standard

L

-function

L (s, π)

vanishes at

s = 1 / 2

, then the theta lift of

π

{S O}_{2 n + 1} (A)

is zero. When

n = 1

, Waldspurger proved that the twisted standard

L

-function

L (s, π, χ)

does not vanish at

s = 1 / 2

for some quadratic automorphic character

χ

A^{\times}

and could use the twisted theta lifting to establish the desired correspondence. But for general

n

, the existence of such a character

χ

is considered extremely difficult to prove.

To circumvent this difficulty, we used the theta lifting in the so-called stable range studied by

L i

[44]. More precisely, for any irreducible genuine representation

π

{M p}_{2 n} (A)

, we consider its (abstract) theta lift

θ^{a b s} (π) = \otimes_{v} θ (π_{v})

{S O}_{2 r + 1} (A)

with

r ≫ 2 n

. Then it follows from the result of

L i

that if

π

occurs in

L_{disc}^{2} ({M p}_{2 n})

, then

θ^{a b s} (π)

occurs in

L_{disc}^{2} ({S O}_{2 r + 1})

. Combining this with the analytic theory of standard

L

-functions, we may deduce Theorem 3.1 from Arthur's endoscopic classification for

{S O}_{2 r + 1}

. Moreover, if

π

is an irreducible summand of the tempered part of

L_{disc}^{2} ({M p}_{2 n})

, then we proved that

m (θ^{a b s} (π)) = m (π)

where

m (\cdot)

denotes the multiplicity in the automorphic discrete spectrum. (We expect that this equality holds for any irreducible summand

π

L_{disc}^{2} ({M p}_{2 n})

.) Using this and describing the local theta lifting between

{M p}_{2 n}

and

{S O}_{2 r + 1}

explicitly, we may deduce Theorem 3.2.

Remark 3.5. When

n = 2

and

ϕ

is nontempered, we proved a similar decomposition of

L_{ϕ}^{2} ({M p}_{4})

in [21]. Note that

π_{η}

is not necessarily irreducible and

ε_{ϕ}

has to be modified in this case.

3.3. Local correspondence

There is a local analog of the above correspondence, called the local Shimura correspondence. For simplicity, we only consider the

p

-adic case and write

F

for a finite extension of

Q_{p}

. We are interested in the set

{IrrMp}_{2 n} (F)

of equivalence classes of irreducible genuine representations of the metaplectic group

{Mp}_{2 n} (F)

. Recall that there are precisely two

(2 n + 1)

-dimensional quadratic spaces

V^{+}

and

V^{-}

over

F

with trivial discriminant (up to isometry). Let

SO (V^{+})

and

SO (V^{-})

denote the special orthogonal groups of

V^{+}

and

V^{-}

, respectively. Then the local Shimura correspondence, which was established by Gan-Savin [22] in the

p

-adic case, says that there is a bijection (depending on a choice of a nontrivial additive character

ψ

F

)

θ : Irr {Mp}_{2 n} (F) \to IrrSO (V^{+}) ⊔ IrrSO (V^{-})

given by the local theta lifting. Namely, for any irreducible genuine representation

π

{Mp}_{2 n} (F), θ (π)

is defined as the unique irreducible representation of

SO (V^{ε})

with the unique sign

ε = \pm

such that

{Hom}_{{Mp}_{2 n} (F) \times SO (V^{ε})} (ω^{ε}, π \otimes θ (π)) \neq 0

where

ω^{ε}

is the Weil representation of

{Mp}_{2 n} (F) \times SO (V^{ε})

(depending on

ψ

). Moreover, they proved various natural properties:

$θ$ preserves the square-integrability,
$θ$ preserves the temperedness,
$θ$ is compatible with the theory of $R$ -groups,
$θ$ is compatible with the Langlands classification,

and used

θ

to transfer the local Langlands correspondence from

SO (V^{ε})

{Mp}_{2 n} (F)

. (In particular, this defines the local

L

-packets in the statement of Theorem 3.2.)

Remark 3.6. The local theta lifting has also been described for other reductive dual pairs in terms of the local Langlands correspondence. See

[3, 4, 19]

for recent progress.

As in Section 2.3, the local theta lifting can produce various relations between local analogs of periods. For example, we consider an irreducible genuine square-integrable representation

π

{Mp}_{2 n} (F)

and its formal degree

d (π)

. Recall that

d (π)

is defined as the positive real number for which the Schur orthogonality relation

\int_{{M p}_{2 n} (F)} ⟨ π (g) v_{1}, v_{2} ⟩ \bar{⟨ π (g) v_{3}, v_{4} ⟩} d g = \frac{1}{d (π)} ⟨ v_{1}, v_{3} ⟩ \bar{⟨ v_{2}, v_{4} ⟩}

holds for all

v_{1}, \dots, v_{4} \in π

, where

⟨ \cdot, \cdot ⟩

is an invariant Hermitian inner product on

π

. Note that

d (π)

depends on the choice of a Haar measure

d g

{Mp}_{2 n} (F)

, but we take the measure determined by a Chevalley basis of the Lie algebra of

{Mp}_{2 n} (F)

and a fixed nontrivial additive character

ψ

F

. The above relation suggests that

d (π)

is a generalization of the dimension of an irreducible representation of a compact group, so that it is a fundamental invariant of a representation. Since a matrix coefficient

g \mapsto ⟨ π (g) v_{1}, v_{2} ⟩

is a local analog of an automorphic form, we may also interpret

d (π)^{- 1}

as a local period. Then we proved in [18] that

d (θ (π)) = d (π)

which is a local analog of the Rallis inner product formula [47], by using the doubling seesaw diagram:

Remark 3.7. Recall from Section 2.3 that some automorphic periods can be expressed in terms of special values of

L

-functions. Similarly, formal degrees should be expressed in terms of arithmetic invariants as follows. Let

G

be a connected reductive group over

F

. For simplicity, we assume that

G

is a pure inner form of a quasisplit group and the center of

G

is anisotropic. Let

π

be an irreducible square-integrable representation of

G (F)

. Let

d (π)

denote the formal degree of

π

with respect to the Haar measure on

G (F)

determined by a Chevalley basis of the Lie algebra of the split form of

G

and a fixed nontrivial additive character

ψ

F

. Then the formal degree conjecture [29] says that

d (π) = \frac{\dim η}{| S_{ϕ} |} | γ (0, Ad \circ ϕ, ψ) |

where

$ϕ : L_{F} \to^{L} G$ is the $L$ -parameter (conjecturally) associated to $π$ ,
$ς_{ϕ}$ is the component group of the centralizer of $ϕ$ in $\hat{G}$ ,
$η$ is the irreducible representation of $ς_{ϕ}$ (conjecturally) associated to $π$ ,
Ad is the adjoint representation of $^{L} G$ on its Lie algebra,
$γ (s$ , Ad $\circ ϕ, ψ)$ is the local $γ$ -factor given by

γ (s, Ad \circ ϕ, ψ) = ε (s, Ad \circ ϕ, ψ) \frac{L (1 - s, Ad \circ ϕ)}{L (s, Ad \circ ϕ)}

In [34], we proved this conjecture for (inner forms of)

{S O}_{2 n + 1}

and its analog for

{M p}_{2 n}

by using the main identity of Lapid-Mao [43] and the above relation between formal degrees.

4. GEOMETRIC REALIZATION OF THE JACOUET-LANGLANDS CORRESPONDENCE

Let

π

be an irreducible automorphic representation of

G (A)

and suppose that

π

is cohomological, so that

π

occurs in the cohomology

H^{*} (X, C)

, where

G

is a connected reductive group and

X

is a locally symmetric space for

G

. Then it is natural to ask whether functorial transfers of

π

can be realized geometrically. In this section, we discuss the simplest example, i.e., the Jacquet-Langlands correspondence for

{G L}_{2}

and its inner forms.

Let

F

be a totally real number field. Let

A^{and} A_{f}

denote the rings of adÃ¨les and finite adÃ¨les of

F

, respectively. Let

π

be an irreducible cuspidal automorphic representation of

{G L}_{2} (A)

such that

π_{v}

is the discrete series of weight 2 for all infinite places

v

F

. For simplicity, we assume that the central character of

π

is trivial, the level of

π

is square-free,
and the Hecke eigenvalues of

π

lie in

Q

. Let

B

be a quaternion division algebra over

F

. For each place

v

F

, put

B_{v} = B \otimes_{F} F_{v}

. Let

V_{B}

be the set of infinite places

v

F

such that

B_{v}

is split. Assume that

V_{B} \neq \emptyset

and put

d = | V_{B} |

. We denote by

X_{B}

the Shimura variety for

B^{\times}

(with respect to some neat open compact subgroup

K_{f}

B^{\times} (A_{f})

), which is a

d

-dimensional smooth projective variety over the reflex field

F^{'}

, so that

$F^{'}$ is the number field contained in $C$ such that

Gal (\bar{Q} / F^{'}) = {σ \in Gal (\bar{Q} / Q) ∣ σ V_{B} = V_{B}}

where

\bar{Q}

denotes the algebraic closure of

Q

C

the $C$ -valued points of $X_{B}$ are given by

X_{B} (C) = B^{\times} ∖ {(H^{\pm})}^{d} \times B^{\times} (A_{f}) / K_{f}

where

S^{\pm}

is the union of the upper and lower half-planes.

Now assume that the Jacquet-Langlands transfer

π^{B}

π

B^{\times} (A)

exists, which is the case if and only if

π_{v}

is a discrete series for all

v

at which

B

is ramified, and that

K_{f}

is chosen appropriately so that

\dim {(π_{f}^{B})}^{K_{f}} = 1

. Here

π_{f}^{B}

is the finite component of

π^{B}

and

{(π_{f}^{B})}^{K_{f}}

is the space of

K_{f}

-fixed vectors in

π_{f}^{B}

. Then it follows from Matsushima's formula that

π_{B}

occurs in the cohomology

H^{*} (X_{B}, C)

. More precisely, we consider the rational cohomology

H^{*} (X_{B}, Q)

and its

π

-isotypic component

H^{*} {(X_{B}, Q)}_{π} = {α \in H^{*} (X_{B}, Q) ∣ T_{v} α = χ_{π_{v}} (T_{v}) α for all T_{v} \in H_{v} and almost all v}

where

H_{v} = Q [K_{v} ∖ B_{v}^{\times} / K_{v}]

is the Hecke algebra with respect to the standard maximal compact subgroup

K_{v}

B_{v}^{\times} ≅ {G L}_{2} (F_{v})

and

χ_{π_{v}} : H_{v} \to Q

is the character by which

H_{v}

acts on

π_{v}^{K_{v}}

. Then

H^{*} {(X_{B}, Q)}_{π}

is concentrated in the middle degree

d

and

H^{d} {(X_{B}, Q)}_{π}

is a

2^{d}

-dimensional vector space over

Q

. Moreover, for any prime

ℓ

, the

ℓ

-adic representation of

Gal (\bar{Q} / F^{'})

H^{d} {(X_{B}, Q)}_{π} \otimes_{Q} Q_{ℓ} ≅ H_{e t}^{d} {(X_{B} \times_{F^{'}} \bar{Q}, Q_{ℓ})}_{π}

is given by the so-called tensor induction of

ρ_{π, ℓ}

, where

ρ_{π, ℓ}

is the 2 -dimensional

ℓ

-adic representation associated to

π

. We remark that it depends only on

ρ_{π, ℓ}

and

V_{B}

Suppose that we have two quaternion algebras

B_{1}

and

B_{2}

as above such that

V_{B_{1}} = V_{B_{2}}

Write

X_{1} = X_{B_{1}}

and

X_{2} = X_{B_{2}}

for the corresponding Shimura varieties, which are of the same dimension

d = | V_{B_{1}} | = | V_{B_{2}} |

over the same reflex field

F^{'}

. Then we have

(1) an (abstract) isomorphism

H^{d} {(X_{1}, C)}_{π} ≅ H^{d} {(X_{2}, C)}_{π}

which preserves the Hodge decomposition,

(2) an (abstract) isomorphism

H^{d} {(X_{1}, Q_{ℓ})}_{π} ≅ H^{d} {(X_{2}, Q_{ℓ})}_{π}

ℓ

-adic representations of

Gal (\bar{Q} / F^{'})

for all

ℓ

Conjecturally, these isomorphisms are obtained from a single isomorphism

H^{d} {(X_{1}, Q)}_{π} ≅ H^{d} {(X_{2}, Q)}_{π}

given as follows. By (2) and the KÃ¼nneth formula, the space of

Gal (\bar{Q} / F^{'})

-fixed vectors in

H^{2 d} {(X_{1} \times X_{2}, Q_{ℓ} (d))}_{π \otimes π}

is nonzero. Hence the Tate conjecture predicts the existence of an algebraic cycle

Z \in {C H}^{d} (X_{1} \times X_{2})

which realizes (1) and (2). Namely, let

p_{1}

and

p_{2}

be the two projections

and consider the following map:

\begin{aligned} H^{d} (X_{1}, Q) \overset{p_{1}^{*}}{\to} & H^{d} (X_{1} \times X_{2}, Q) \\ ↓ \cup [Z] \\ H^{3 d} (X_{1} \times X_{2}, Q (d)) \overset{p_{2 *}}{\to} H^{d} (X_{2}, Q) . \end{aligned}

Then it induces an isomorphism

ι_{Z} : H^{d} {(X_{1}, Q)}_{π} \to H^{d} {(X_{2}, Q)}_{π}

such that

$ι_{Z} \otimes {id}_{C}$ preserves the Hodge decomposition,
$ι_{Z} \otimes {id}_{Q_{ℓ}}$ is $Gal (\bar{Q} / F^{'})$ -equivariant for all $ℓ$ .

When

d = 1

, the existence of

Z

in fact follows from the result of Faltings [16]. But for general

d

, this remains an open problem. On the other hand, noting that the Hodge conjecture reduces it to finding a Hodge cycle on

X_{1} \times X_{2}

, i.e., an element in

H^{2 d} (X_{1} \times X_{2}, Q) \cap H^{d, d} (X_{1} \times X_{2})

we gave the following evidence.

Theorem 4.1 ([35]). Assume that

B_{1}

and

B_{2}

are ramified at some infinite place

v

F

. Then there exists a Hodge cycle

ξ

X_{1} \times X_{2}

which induces an isomorphism

ι_{ξ} : H^{d} {(X_{1}, Q)}_{π} \to H^{d} {(X_{2}, Q)}_{π}

such that

$ι ξ \otimes {i d}_{C}$ preserves the Hodge decomposition,
$ι_{ξ} \otimes {id}_{Q_{ℓ}}$ is $Gal (\bar{Q} / F^{'})$ -equivariant for all $ℓ$ .

Our proof proceeds as follows. First, we choose an ambient variety

X

equipped with an embedding

j : X_{1} \times X_{2} ↪ X

. Then we construct a class

Ξ \in H^{d, d} (X)

such that the

(π \otimes π)

-isotypic component

{(j^{*} Ξ)}_{π \otimes π}

of the pullback

j^{*} Ξ \in H^{d, d} (X_{1} \times X_{2})

is nonzero. Finally, we modify

Ξ

in such a way that

Ξ

lies in

H^{2 d} (X, Q)

and

ξ = {(j^{*} Ξ)}_{π \otimes π}

is the desired Hodge cycle.

More precisely, fix a totally imaginary quadratic extension

E

F

which embeds into

B_{1}

and

B_{2}

. For

i = 1, 2

, let

V_{i} = B_{i}

be the 2-dimensional Hermitian space over

E

such that

G U (V_{i}) = (B_{i}^{\times} \times E^{\times}) / F^{\times}

Then we may replace

X_{i}

by the Shimura variety for

G U (V_{i})

. Consider the 4-dimensional Hermitian space

V = V_{1} \oplus V_{2}

over

E

and put

G = G U (V)

. Note that if we write

v_{1}, \dots, v_{d}

(resp.

v_{d + 1}, \dots, v_{[F : Q]}

) for the infinite places of

F

at which

B_{1}

and

B_{2}

are split (resp. ramified), then we have

G (F_{v_{i}}) = {\begin{cases} GU (2, 2) & if i \leq d \\ GU (4) & if i > d \end{cases}

Put

G_{\infty} = G (F \otimes_{Q} R)

and let

g_{\infty}

denote the complexified Lie algebra of

G_{\infty}

. Let

K_{\infty}

be the standard maximal connected compact modulo center subgroup of

G_{\infty}

. Let

X

be the Shimura variety for

G

(with respect to some neat open compact subgroup

K_{f}

G (A_{f})

), which is equipped with the embedding

j : X_{1} \times X_{2} ↪ X

induced by the natural embedding

G (U (V_{1}) \times U (V_{2})) ↪ G U (V)

where the left-hand side is the subgroup of

GU (V_{1}) \times G U (V_{2})

which consists of elements with the same similitude factor. Then Matsushima's formula says that

H^{*} (X, C) ≅ ⨁_{σ} m (σ) H^{*} (g_{\infty}, K_{\infty}; σ_{\infty}) \otimes σ_{f}^{K_{f}}

where

$σ$ runs over irreducible unitary representations of $G (A)$ ,
$σ_{\infty}$ and $σ_{f}$ are the infinite and finite components of $σ$ , respectively,
$m (σ)$ is the multiplicity of $σ$ in the automorphic discrete spectrum of $G$ ,
$H^{*} (g_{\infty}, K_{\infty}; σ_{\infty})$ is the relative Lie algebra cohomology,
$σ_{f}^{K_{f}}$ is the space of $K_{f}$ -fixed vectors in $σ_{f}$ .

Hence, to construct a class

Ξ

as above, we need to find an irreducible automorphic representation

σ

G (A)

which satisfies the following properties:
(a) To achieve the condition

Ξ \in H^{d, d} (X)

, we require that

H^{d, d} (g_{\infty}, K_{\infty}; σ_{\infty}) \neq 0

If this is the case, then it follows from the result of Vogan-Zuckerman [52] that

σ_{v_{i}}

(restricted to

U (V) (F_{v_{i}}))

is equal to

{\begin{cases} 1 or A_{q} & if i \leq d \\ 1 & if i > d \end{cases}

Here

1

denotes the trivial representation and

A_{q}

is the cohomological representation of

U (2, 2)

associated to the

θ

-stable parabolic subalgebra q with Levi component

u (1, 1) \oplus u (1, 1)

. We further require that

σ_{v_{i}} = A_{q}

i \leq d

in order not to make

σ

1-dimensional.

(b) To achieve the condition

{(j^{*} Ξ)}_{π \otimes π} \neq 0

, we require the nonvanishing of the automorphic period

σ \otimes \bar{(π^{B_{1}} \otimes π^{B_{2}})} \to C

For (a), we use the following variant of the theta lifting from

{SL}_{2}

SO (4, 2) \sim U (2, 2)

S O (6) \sim U (4)

, where

\sim

denotes an isogeny. Let

B

be the quaternion algebra over

F

such that

B = B_{1} \cdot B_{2}

in the Brauer group, so that

B

is split at all infinite places of

F

. We may regard

V = \land^{2} V

as a 3-dimensional skew-Hermitian space over

B

such that

G U (V)^{0} / F^{\times} ≅ G U (V) / E^{\times}

Let

W = B

be the 1-dimensional Hermitian space over

B

such that

G U (W) = B^{\times}

Then any

σ

as in (a) with trivial central character is a theta lift of an irreducible cuspidal automorphic representation

τ

GU (W) (A)

such that

τ_{v}

is the discrete series of weight 3 for all infinite places

v

F

. For (b), we can easily find the corresponding

τ

by using the following seesaw diagram:

Here

V = V_{1} \oplus V_{2}

is a decomposition into 1- and 2-dimensional skew-Hermitian spaces over

B

such that

GU {(V_{1})}^{0} = E^{\times}, G U {(V_{2})}^{0} = (B_{1}^{\times} \times B_{2}^{\times}) / F^{\times}

and

G {(U (V_{1}) \times U (V_{2}))}^{0} / F^{\times} ≅ G (U (V_{1}) \times U (V_{2})) / E^{\times}

For simplicity, we further assume that the Hecke eigenvalues of

σ

lie in

Q

. Thus we obtain a class

Ξ \in H^{d, d} (X)

such that

ξ = {(j^{*} Ξ)}_{π \otimes π}

induces an isomorphism

H^{d} {(X_{1}, C)}_{π} ≅ H^{d} {(X_{2}, C)}_{π}

To be precise, we need to use the theta lifting valued in cohomology developed by KudlaMillson [40,41]. On the other hand, we can determine the near equivalence class of

σ

and prove that

H^{2 d} (X, C)_{σ} \subset H^{d, d} (X)

Hence we can modify

Ξ

in such a way that

Ξ

lies in

H^{2 d} (X, Q)_{σ}

, so that it is a Hodge cycle. Finally, it follows from the result of Kisin-Shin-Zhu [38] that

H^{2 d} {(X, Q_{ℓ})}_{σ} ≅ Q_{ℓ} (- d)^{m}

for some positive integer

m

, from which Theorem 4.1 follows immediately.

Remark 4.2. It is desirable to upgrade

ξ

to an absolute Hodge cycle in the sense of Deligne [14], but this remains an open problem.

ACKNOWLEDGMENTS

The author would like to thank Tamotsu Ikeda for his constant support and encouragement. The author would also like to thank Wee Teck Gan, Erez Lapid, and Kartik Prasanna for their generosity in sharing their ideas and numerous valuable discussions over the years.

FUNDING

This work was partially supported by JSPS KAKENHI Grant Number 19H01781.

REFERENCES

[1] J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Amer. Math. Soc. Colloq. Publ. 61, American Mathematical Society, Providence, RI, 2013.

[2] H. Atobe, The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case. Math. Ann. 371 (2018), no. 1-2, 225-295.

[3] H. Atobe and W. T. Gan, Local theta correspondence of tempered representations and Langlands parameters. Invent. Math. 210 (2017), no. 2, 341-415.

[4] P. BakiÄ‡ and M. Hanzer, Theta correspondence for

p

-adic dual pairs of type I. J. Reine Angew. Math. 776 (2021), 63-117.

[5] R. Beuzart-Plessis, Expression d'un facteur epsilon de paire par une formule intÃ©grale. Canad. J. Math. 66 (2014), no. 5, 993-1049.

[6] R. Beuzart-Plessis, Endoscopie et conjecture locale raffinÃ©e de Gan-Gross-Prasad pour les groupes unitaires. Compos. Math. 151 (2015), no. 7, 1309-1371.

[7] R. Beuzart-Plessis, La conjecture locale de Gross-Prasad pour les reprÃ©sentations tempÃ©rÃ©es des groupes unitaires. MÃ©m. Soc. Math. Fr. (N.S.) 149 (2016).

[8] R. Beuzart-Plessis, Plancherel formula for

{G L}_{n} (F) ∖ {G L}_{n} (E)

and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups. Invent. Math. 225 (2021), no. 1, 159-297.

[9] R. Beuzart-Plessis, Comparison of local relative characters and the Ichino-Ikeda conjecture for unitary groups. J. Inst. Math. Jussieu 20 (2021), no. 6, 1803-1854.

[10] R. Beuzart-Plessis, P.-H. Chaudouard, and M. Zydor, The global Gan-GrossPrasad conjecture for unitary groups: the endoscopic case. 2020,

arXiv 2007.05601 .

[11] R. Beuzart-Plessis, Y. Liu, W. Zhang, and X. Zhu, Isolation of cuspidal spectrum, with application to the Gan-Gross-Prasad conjecture. Ann. of Math. (2) 194 (2021), no. 2, 519-584.

[12] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to

{G L}_{N}

. Publ. Math. Inst. Hautes Ã‰tudes Sci. 93 (2001), 5-30.

[13] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups. Publ. Math. Inst. Hautes Ã‰tudes Sci. 99 (2004), 163-233.

[14] P. Deligne, Hodge cycles on abelian varieties (Notes by J. S. Milne). In Hodge cycles, motives, and Shimura varieties, pp. 9-100, Lecture Notes in Math. 900, Springer, Berlin-New York, 1982.

[15] M. Eichler, Quadratische Formen und Modulfunktionen. Acta Arith. 4 (1958), 217-239.

[16] G. Faltings, EndlichkeitssÃ¤tze fÃ¼r abelsche VarietÃ¤ten Ã¼ber ZahlkÃ¶rpern. Invent. Math. 73 (1983), no. 3, 349-366.

[17] W. T. Gan, B. H. Gross, and D. Prasad, Symplectic local root numbers, central critical

L

[18] W. T. Gan and A. Ichino, Formal degrees and local theta correspondence. Invent. Math. 195 (2014), no. 3, 509-672.

[19] W. T. Gan and A. Ichino, The Gross-Prasad conjecture and local theta correspondence. Invent. Math. 206 (2016), no. 3, 705-799.

[20] W. T. Gan and A. Ichino, The Shimura-Waldspurger correspondence for

{M p}_{2 n}

. Ann. of Math. (2) 188 (2018), no. 3, 965-1016.

[21] W. T. Gan and A. Ichino, The automorphic discrete spectrum of

{M p}_{4}

. Int. Math. Res. Not. IMRN 3 (2021), 1603-1677.

[22] W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compos. Math. 148 (2012), no. 6, 1655-1694.

[23] W. T. Gan and S. Takeda, A proof of the Howe duality conjecture. J. Amer. Math. Soc. 29 (2016), no. 2, 473-493.

[24] D. Ginzburg, S. Rallis, and D. Soudry, The descent map from automorphic representations of GL(n) to classical groups. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.

[25] B. H. Gross and D. Prasad, On the decomposition of a representation of

{S O}_{n}

when restricted to

{S O}_{n - 1}

. Canad. J. Math. 44 (1992), no. 5, 974-1002.

[26] B. H. Gross and D. Prasad, On irreducible representations of

{S O}_{2 n + 1} \times {S O}_{2 m}

. Canad. J. Math. 46 (1994), no. 5, 930-950.

[27] M. Harris and S. S. Kudla, The central critical value of a triple product L-function. Ann. of Math. (2) 133 (1991), no. 3, 605-672.

[28] E. Hecke, Neuere Fortschritte in der Theorie der elliptischen Modulfunktionen. In Comptes Rendus du CongrÃ¨s International des MathÃ©maticiens, Tome I (Oslo, 1936), pp. 140-156, A. W. BrÃ¸ggers Boktrykkeri A/S, Oslo, 1937.

[29] K. Hiraga, A. Ichino, and T. Ikeda, Formal degrees and adjoint

γ

-factors. J. Amer. Math. Soc. 21 (2008), no. 1, 283-304.

[30] R. Howe,

θ

-series and invariant theory. In Automorphic forms, representations and L-functions, Part 1 (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), pp. 275-285, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, RI, 1979.

[31] R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535-552.

[32] A. Ichino, Trilinear forms and the central values of triple product

L

-functions. Duke Math. J. 145 (2008), no. 2, 281-307.

[33] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal. 19 (2010), no. 5, 1378-1425.

[34] A. Ichino, E. Lapid, and Z. Mao, On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups. Duke Math. J. 166 (2017), no. 7, 1301-1348.

[35] A. Ichino and K. Prasanna, Hodge classes and the Jacquet-Langlands correspondence. 2018, arXiv:1806.10563.

[36] A. Ichino and K. Prasanna, Periods of quaternionic Shimura varieties. I. Contemp. Math. 762, American Mathematical Society, Providence, RI, 2021.

[37] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2). Lecture Notes in Math. 114, Springer, Berlin-New York, 1970.

[38] M. Kisin, S. W. Shin, and Y. Zhu, The stable trace formula for certain Shimura varieties of abelian type. 2021, arXiv:2110.05381.

[39] S. S. Kudla, Seesaw dual reductive pairs. In Automorphic forms of several variables (Katata, 1983), pp. 244-268, Progr. Math. 46, BirkhÃ¤user, Boston, MA, 1984.

[40] S. S. Kudla and J. J. Millson, The theta correspondence and harmonic forms. I. Math. Ann. 274 (1986), no. 3, 353-378.

[41] S. S. Kudla and J. J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Publ. Math. Inst. Hautes Ã‰tudes Sci. 71 (1990), 121-172.

[42] R. P. Langlands, Problems in the theory of automorphic forms. In Lectures in modern analysis and applications, III, pp. 18-61, Lecture Notes in Math. 170, Springer, Berlin, 1970 .

[43] E. Lapid and Z. Mao, On an analogue of the Ichino-Ikeda conjecture for Whittaker coefficients on the metaplectic group. Algebra Number Theory 11 (2017), no. 3, 713-765.

[44] J.-S. Li, Automorphic forms with degenerate Fourier coefficients. Amer. J. Math. 119 (1997), no. 3, 523-578.

[45] C. MÅ“glin and J.-L. Waldspurger, La conjecture locale de Gross-Prasad pour les groupes spÃ©ciaux orthogonaux: le cas gÃ©nÃ©ral. AstÃ©risque 347 (2012), 167-216.

[46] S. Niwa, Modular forms of half integral weight and the integral of certain thetafunctions. Nagoya Math. J. 56 (1975), 147-161.

[47] S. Rallis, Injectivity properties of liftings associated to Weil representations. Compos. Math. 52 (1984), no. 2, 139-169.

[48] H. Shimizu, On zeta functions of quaternion algebras. Ann. of Math. (2) 81 (1965), 166-193.

[49] H. Shimizu, Theta series and automorphic forms on

{G L}_{2}

. J. Math. Soc. Japan 24 (1972), 638-683.

[50] G. Shimura, On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440-481.

[51] T. Shintani, On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975), 83-126.

[52] D. A. Jr. Vogan and G. J. Zuckerman, Unitary representations with nonzero cohomology. Compos. Math. 53 (1984), no. 1, 51-90.

[53] J.-L. Waldspurger, Correspondance de Shimura. J. Math. Pures Appl. (9) 59 (1980), no. 1, 1-132.

[54] J.-L. Waldspurger, Sur les valeurs de certaines fonctions

L

p

-adique,

p \neq 2

. In Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), pp. 267-324, Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990.

[56] J.-L. Waldspurger, Correspondances de Shimura et quaternions. Forum Math. 3 (1991), no. 3, 219-307.

[57] J.-L. Waldspurger, Une formule intÃ©grale reliÃ©e Ã la conjecture locale de GrossPrasad. Compos. Math. 146 (2010), no. 5, 1180-1290.

[58] J.-L. Waldspurger, Calcul d'une valeur d'un facteur

ϵ

par une formule intÃ©grale. AstÃ©risque 347 (2012), 1-102.

[59] J.-L. Waldspurger, La conjecture locale de Gross-Prasad pour les reprÃ©sentations tempÃ©rÃ©es des groupes spÃ©ciaux orthogonaux. AstÃ©risque 347 (2012), 103-165.

[60] J.-L. Waldspurger, Une formule intÃ©grale reliÃ©e Ã la conjecture locale de GrossPrasad, 2e partie: extension aux reprÃ©sentations tempÃ©rÃ©es. AstÃ©risque 346 (2012),

171 - 312

143 - 211

[62] H. Xue, On the global Gan-Gross-Prasad conjecture for unitary groups: approximating smooth transfer of Jacquet-Rallis. J. Reine Angew. Math. 756 (2019),

65 - 100

[63] W. Zhang, Automorphic period and the central value of Rankin-Selberg L-function. J. Amer. Math. Soc. 27 (2014), no. 2, 541-612.

[64] W. Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups. Ann. of Math. (2) 180 (2014), no. 3, 971-1049.

ATSUSHI ICHINO

Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan, ichino @ math.kyoto-u.ac.jp

RATIONAL

APPROXIMATIONS

OF IRRATIONAL NUMBERS

DIMITRIS KOUKOULOPOULOS

ABSTRACT

Given quantities

Δ_{1}, Δ_{2}, \dots ⩾ 0

, a fundamental problem in Diophantine approximation is to understand which irrational numbers

x

have infinitely many reduced rational approximations

a / q

such that

| x - a / q | < Δ_{q}

. Depending on the choice of

Δ_{q}

and of

x

, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a "metric" point of view, the question is governed by a simple zeroone law: writing

φ

for Euler's totient function, we either have

\sum_{q = 1}^{\infty} φ (q) Δ_{q} = \infty

and then almost all irrational numbers (in the Lebesgue sense) are approximable, or

\sum_{q = 1}^{\infty} φ (q) Δ_{q} < \infty

and almost no irrationals are approximable. We will present the history of the Duffin-Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos-Maynard that settled it.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11J83; Secondary 05C40

KEYWORDS

Diophantine approximation, Duffin-Schaeffer conjecture, Metric Number Theory, compression arguments, density increment, graph theory

1. DIOPHANTINE APPROXIMATION

Let

x

be an irrational number. In many settings, practical and theoretical, it is important to find fractions

a / q

of small numerator and denominator that approximate it well. This fundamental question lies at the core of the field of Diophantine approximation.

1.1. First principles

The "high-school way" of approximating

x

is to use its decimal expansion. This approach produces fractions

a / 10^{n}

such that

| x - a / 10^{n} | \approx 10^{- n}

typically. However, the error can be made much smaller if we allow more general denominators [14, THEOREM 2.1].

Theorem 1.1. If

x \in R ∖ Q

, then

| x - a / q | < q^{- 2}

for infinitely many pairs

(a, q) \in Z \times N

Dirichlet (c. 1840) gave a short and clever proof of this theorem. However, his argument is nonconstructive because it uses the pigeonhole principle. This gap is filled by the theory of continued fractions (which actually precedes Dirichlet's proof).

Given any

x \in R ∖ Q

, we may write

x = n_{0} + r_{0} \approx n_{0}

, where

n_{0} = ⌊ x ⌋

is the integer part of

x

and

r_{0} = {x}

is its fractional part. We then let

n_{1} = ⌊ 1 / r_{0} ⌋

and

r_{1} = {1 / r_{0}}

, so that

x = n_{0} + 1 / (n_{1} + r_{1}) \approx n_{0} + 1 / n_{1}

. If we repeat this process

j - 1

more times, we find that

\begin{matrix} (1.1) & x \approx n_{0} + \frac{1}{n_{1} + \frac{1}{\dots + \frac{1}{n_{j}}}} with n_{i} = ⌊ \frac{1}{r_{i - 1}} ⌋, r_{i} = {\frac{1}{r_{i - 1}}} for i = 1, \dots, j . \end{matrix}

If we write this fraction as

a_{j} / q_{j}

in reduced form, then a calculation reveals that

\begin{aligned} (1.2) & a_{j} = n_{j} a_{j - 1} + a_{j - 2} (j ⩾ 2), a_{1} = n_{0} n_{1} + 1, a_{0} = n_{0} \\ q_{j} = n_{j} q_{j - 1} + q_{j - 2} (j ⩾ 2), q_{1} = n_{1}, q_{0} = 1 \end{aligned}

When

j \to \infty

, the right-hand side of (1.1), often denoted by

[n_{0}; n_{1}, \dots, n_{j}]

, converges to

x

. The resulting representation of

x

is called its continued fraction expansion. The quotients

a_{j} / q_{j}

are called the convergents of this expansion and they have remarkable properties [17]. We list some of them below, with the first giving a constructive proof of Theorem 1.1.

Theorem 1.2. Assume the above set-up and notations.

(a) For each

j ⩾ 0

, we have

1 / (2 q_{j} q_{j + 1}) ⩽ | x - a_{j} / q_{j} | ⩽ 1 / (q_{j} q_{j + 1})

(b) For each

j ⩾ 0

, we have

| x - a_{j} / q_{j} | = min {| x - a / q | : 1 ⩽ q ⩽ q_{j}}

| x - a / q | < 1 / (2 q^{2})

with

a

and

q

coprime, then

a / q = a_{j} / q_{j}

for some

j ⩾ 0

1.2. Improving Dirichlet's approximation theorem

It is natural to ask when a qualitative improvement of Theorem 1.1 exists. Inverting this question leads us to the following definition: we say that a real number

x

is badly approximable if there is

c = c (x) > 0

such that

| x - a / q | ⩾ c q^{- 2}

for all

(a, q) \in Z \times N

We can characterize approximable numbers in terms of their continued fraction expansion. Indeed, Theorem 1.2(a) and relation (1.2) imply that

1 / 4 ⩽ n_{j + 1} q_{j}^{2} | x - a_{j} / q_{j} | ⩽ 1

Hence, together with Theorem 1.2(c), this implies that

x

is badly approximable if and only if the sequence

{(n_{j})}_{j = 0}^{\infty}

is bounded. Famously, Lagrange proved that the quadratic irrational numbers are in one-to-one correspondence with the continued fractions that are eventually periodic [17,

$ 10]

. In particular, all such numbers are badly approximable.

A related concept to badly approximable numbers is the irrationality measure. For each

x \in R

, we define it to be

μ (x) := sup {v ⩾ 0 : 0 < | x - a / q | < q^{- ν} for infinitely many pairs (a, q) \in Z \times N}

Note that

μ (x) = 1

x \in Q

, whereas

μ (x) ⩾ 2

x \in R ∖ Q

by Theorem 1.1. Moreover,

μ (x) = 2

x

is badly approximable. In particular,

μ (x) = 2

for all quadratic irrationals

x

. Remarkably, Roth [25] proved that

μ (x) = 2

for all algebraic irrational numbers

x

Determining the irrationality measure of various famous transcendental constants is often very hard. We do know that

μ (e) = 2

, where

e

denotes Euler's constant. However, determining

μ (π)

is a famous open problem. Towards it, Zeilberger and Zudilin [29] proved that

μ (π) ⩽ 7.10320533 \dots

It is widely believed that

μ (π) = 2

Instead of trying to reduce the error term in Dirichlet's approximation theorem, we often require a different type of improvement: restricting the denominators

q

to lie in some special set

S

. The theory of continued fractions is of limited use for such problems because the denominators it produces satisfy rigid recursive relations (cf. (1.2)).

For rational approximation with prime or square denominators, the best results at the moment are due to MatomÃ¤ki [21] and Zaharescu [28], respectively.

Theorem 1.3 (MatomÃ¤ki (2009)). Let

x

be an irrational number and let

ε > 0

. There are infinitely many integers

a

and prime numbers

p

such that

| x - a / p | < p^{- 4 / 3 + ε}

Theorem 1.4 (Zaharescu (1995)). Let

x

be an irrational number and let

ε > 0

. There are infinitely many pairs

(a, q) \in Z \times N

such that

| x - a / q^{2} | < q^{- 8 / 3 + ε}

Two important open problems are to show that Theorems 1.3 and 1.4 remain true even if we replace the constants

4 / 3

and

8 / 3

by 2 and 3, respectively.

2. METRIC DIOPHANTINE APPROXIMATION

Unable to answer simple questions about the rational approximations of specific numbers, a lot of research adopted a more statistical point of view. For example, given

M > 2

, what proportion of real numbers have irrationality measure

⩾ M

? This new perspective gives rise to the theory of metric Diophantine approximation, which has a much more analytic and probabilistic flavor than the classical theory of Diophantine approximation. As we will see, the ability to ignore small pathological sets of numbers leads to a much more robust theory that provides simple and satisfactory answers to very general questions.

In order to give precise meaning to the word "proportion," we shall endow

R

with a measure. Here, we will mainly use the Lebesgue measure (denoted by "meas").

2.1. The theorems of Khinchin and JarnÃk-Besicovitch

The foundational result in the field of metric Diophantine approximation was proven by Khinchin in his seminal 1924 paper [18]. It is a rather general result: given a sequence

Δ_{1}, Δ_{2}, \dots, ⩾ 0

of "permissible margins of error," we wish to determine for which real numbers

x

there are infinitely many pairs

(a, q) \in Z \times N

such that

| x - a / q | < Δ_{q}

. Clearly, if

x

has this property, so does

x + 1

. Hence, we may focus on studying

\begin{matrix} (2.1) & A := {x \in [0, 1] : | x - a / q | < Δ_{q} for infinitely many pairs (a, q) \in Z \times N} \end{matrix}

Theorem 2.1 (Khinchin (1924)). Let

Δ_{1}, Δ_{2}, \dots ⩾ 0

and let

A

be defined as in (2.1).

(a) If

\sum_{q = 1}^{\infty} q Δ_{q} < \infty

, then meas

(A) = 0

(b) If

\sum_{q = 1}^{\infty} q Δ_{q} = \infty

and the sequence

{(q^{2} Δ_{q})}_{q = 1}^{\infty}

is decreasing, then

meas (A) = 1

Corollary 2.2. For almost all

x \in R

, we have

| x - a / q | ⩽ 1 / (q^{2} \log q)

for infinitely many

(a, q) \in Z \times N

. On the other hand, if

c > 1

is fixed, then for almost every

x \in R

, the inequality

| x - a / q | ⩽ 1 / (q^{2} \log^{c} q)

admits only finitely many solutions

(a, q) \in Z \times N

In particular, Corollary 2.2 implies that the set of badly approximable numbers has null Lebesgue measure. On the other hand, it also says that almost all real numbers have irrationality measure equal to 2 . This last result is the main motivation behind the conjecture that

μ (π) = 2

: we expect

π

to behave like a "typical" real number.

Naturally, the fact that

W_{M} := {x \in R : μ (x) ⩾ M}

has null Lebesgue measure for

M > 2

raises the question of determining its Hausdorff dimension (denoted by

\dim (W_{M})

). JarnÃk [16] and Besicovitch [5] answered this question independently of each other.

Theorem 2.3 (JarnÃk (1928), Besicovitch (1934)). We have

\dim (W_{M}) = 2 / M

for all

M ⩾ 2

2.2. Generalizing Khinchin's theorem

Following the publication of Khinchin's theorem, research focused on weakening the assumption that

q^{2} Δ_{q} ↘

in part (b). Importantly, doing so would open the door to understanding rational approximations using only a restricted set of denominators. Indeed, if

q^{2} Δ_{q} ↘

, then either

Δ_{q} > 0

for all

q

, or there is

q_{0}

such that

Δ_{q} = 0

for all

q ⩾ q_{0}

. The second case is trivial, since it implies

A = \emptyset

. So, if we wish to understand Diophantine approximation with a restricted set of denominators

S

(which would require

Δ_{q} = 0

for

q \notin S

), then we must prove a version of Theorem 2.1(b) without the assumption that

q^{2} Δ_{q} ↘

In order to understand better the forces at play here, it is useful to recast Khinchin's theorem in probabilistic terms. For each

q

, let us define the set

\begin{aligned} A_{q} & := {x \in [0, 1] : there is a \in Z such that | x - \frac{a}{q} | < Δ_{q}} \\ = [0, 1] \cap ⋃_{0 ⩽ a ⩽ q} (\frac{a}{q} - Δ_{q}, \frac{a}{q} + Δ_{q}) \end{aligned}

Then

A = {x \in [0, 1] : x \in A_{q}

infinitely often

}

, which we often write as

A = lim sup_{q \to \infty} A_{q}

. We may thus view

A

as the event that for a number chosen uniformly at random from

[0, 1]

, an infinite number of the events

A_{1}, A_{2}, \dots

occur. A classical result from probability theory due to Borel and Cantelli [14, LEMMAS 1.2 E 1.3] studies precisely this kind of questions.

Theorem 2.4. Let

(Ω, F, P)

be a probability space, let

E_{1}, E_{2}, \dots

be events in that space, and let

E = lim sup \sin_{j \to \infty} E_{j}

be the event that infinitely many of the

E_{j}

's occur.

(a) (The first Borel-Cantelli lemma) If

\sum_{j = 1}^{\infty} P (E_{j}) < \infty

, then

P (E) = 0

(b) (The second Borel-Cantelli lemma) If

\sum_{j = 1}^{\infty} P (E_{j}) = \infty

and the events

E_{1}, E_{2}, \dots

are mutually independent, then

P (E) = 1

Remark. Let

N

be the random variable that counts how many of the events

E_{1}, E_{2}, \dots

occur. We have

E [N] = \sum_{j = 1}^{\infty} P (E_{j})

. Hence, Theorem 2.4 says that, under certain assumptions,

N = \infty

almost surely if, and only if,

E [N] = \infty

To use the above result in the setup of Khinchin's theorem, we let

Ω = [0, 1]

and equip it with the Lebesgue measure as its probability measure. The relevant events

E_{j}

are the sets

A_{q}

. Notice that if

Δ_{q} > 1 / (2 q)

, then

A_{q} = [0, 1]

, in which case

A_{q}

occurs immediately for all

x \in [0, 1]

. In order to avoid these trivial events, we will assume from now on that

\begin{matrix} (2.3) & Δ_{q} ⩽ 1 / (2 q) for all q ⩾ 1, whence meas (A_{q}) = 2 q Δ_{q} \end{matrix}

In particular, we see that part (a) of Khinchin's theorem is a direct consequence of the first Borel-Cantelli lemma. On the other hand, the second Borel-Cantelli lemma relies crucially on the assumption that the events

E_{j}

are independent of each other, something that fails generically for the events

A_{q}

. However, there are variations of the second BorelCantelli lemma, where the assumption of independence can be replaced by weaker quasiindependence conditions on the relevant events (cf. Section 3.1). From this perspective, part (b) of Khinchin's theorem can be seen as saying that the condition that the sequence

{(q^{2} Δ_{q})}_{q = 1}^{\infty}

is decreasing guarantees enough approximate independence between the events

A_{q}

so that the conclusion of the second Borel-Cantelli lemma remains valid.

In 1941, Duffin and Schaeffer published a seminal paper [8] that studied precisely what is the right way to generalize Khinchin's theorem so that the simple zero-one law of Borel-Cantelli holds. Their starting point was the simple observation that certain choices of the quantities

Δ_{q}

create many dependencies between the sets

A_{q}

, thus rendering many of the denominators

q

redundant. Indeed, note, for example, that if

Δ_{3} = Δ_{15}

, then

A_{3} \subseteq A_{15}

because each fraction with denominator 3 can also be written as a fraction with denominator 15. By exploiting this simple idea, Duffin and Schaeffer proved the following result:

Proposition 2.5. There are

Δ_{1}, Δ_{2}, \dots ⩾ 0

such that

\sum_{q = 1}^{\infty} q Δ_{q} = \infty

and yet meas(A)

= 0

. Proof. Let

p_{1} < p_{2} < \dots

be the primes in increasing order, let

q_{j} = p_{1} \dots p_{j}

, and let

S_{j} = {d p_{j} : d ∣ q_{j - 1}}

. We then set

Δ_{q} = {(q_{j} j \log^{2} j)}^{- 1}

q \in S_{j}

for some

j ⩾ 2

; otherwise, we set

Δ_{q} = 0

. We claim that this choice satisfies the needed conditions.

Since

A_{q} \subseteq A_{q_{j}}

for all

q \in S_{j}

, we have

A = lim sup \sin_{j \to \infty} A_{q_{j}}

. In addition, since

\sum_{j = 1}^{\infty} q_{j} Δ_{q_{j}} < \infty

, we have meas

(lim sup_{j \to \infty} A_{q_{j}}) = 0

by Theorem 2.1(a). Hence, meas

(A) = 0

, as needed. On the other hand, we have that

\sum_{q ⩾ 1} q Δ_{q} = \sum_{j ⩾ 2} \sum_{d ∣ q_{j - 1}} d p_{j} \cdot \frac{1}{q_{j} j \log^{2} j} = \sum_{j ⩾ 2} \frac{1}{j \log^{2} j} \prod_{i ⩽ j - 1} (1 + \frac{1}{p_{i}})

By the Prime Number Theorem [19, THEOREM 8.1], the last product is

⩾ c \log j

for some absolute constant

c > 0

. Consequently,

\sum_{q = 1}^{\infty} q Δ_{q} = \infty

, as claimed.

In order to avoid the above kind of counterexamples to the generalized Khinchin theorem, Duffin and Schaeffer were naturally led to consider a modified setup, where only reduced fractions are used as approximations. They thus defined

\begin{matrix} (2.4) & A^{*} := {x \in [0, 1] : | x - a / q | < Δ_{q} for infinitely many reduced fractions a / q} \end{matrix}

We may write

A^{*}

as the lim sup of the sets

\begin{matrix} (2.5) & A_{q}^{*} := [0, 1] \cap ⋃_{\begin{matrix} 0 ⩽ a ⩽ q \\ \gcd (a, q) = 1 \end{matrix}} (\frac{a}{q} - Δ_{q}, \frac{a}{q} + Δ_{q}) \end{matrix}

Assuming that (2.3) holds, we readily find that

meas (A_{q}^{*}) = 2 φ (q) Δ_{q}

where

φ (q) := # {1 ⩽ a ⩽ q : \gcd (a, q) = 1}

is Euler's totient function. They then conjectured that the sets

A_{q}^{*}

have enough mutual quasiindependence so that a simple zero-one law holds, as per the Borel-Cantelli lemmas.

The Duffin-Schaeffer conjecture. Let

Δ_{1}, Δ_{2}, \dots ⩾ 0

and let

A^{*}

be defined as in (2.4).

(a) If

\sum_{q = 1}^{\infty} φ (q) Δ_{q} < \infty

, then meas

(A^{*}) = 0

(b) If

\sum_{q = 1}^{\infty} φ (q) Δ_{q} = \infty

, then meas

(A^{*}) = 1

Of course, part (a) follows from Theorem 2.4(a); the main difficulty is to prove (b).

The Duffin-Schaeffer conjecture is strikingly simple and general. Nonetheless, it does not answer our original question: what is the correct generalization of Khinchin's theorem, where we may use nonreduced fractions? This gap was filled by Catlin [7].

Catlin's conjecture. Let

Δ_{1}, Δ_{2}, \dots ⩾ 0

, let

Δ_{q}^{'} = sup_{m ⩾ 1} Δ_{q m}

, and let

A

be as in (2.1).

(a) If

\sum_{q = 1}^{\infty} φ (q) Δ_{q}^{'} < \infty

, then

meas (A) = 0

(b) If

\sum_{q = 1}^{\infty} φ (q) Δ_{q}^{'} = \infty

, then meas

(A) = 1

As Catlin noticed, his conjecture is a direct corollary of that by Duffin and Schaeffer. Indeed, let us consider the set

A^{'} = {x \in [0, 1] : | x - a / q | < Δ_{q}^{'} for infinitely many reduced fractions a / q}

This is the set

A^{*}

with the quantities

Δ_{q}

replaced by

Δ_{q}^{'}

, so we may apply the DuffinSchaeffer conjecture to it. In addition, it is straightforward to check that

\begin{matrix} (2.6) & A ∖ Q = A^{'} ∖ Q \end{matrix}

when

Δ_{q} \to 0

. This settles Catlin's conjecture in this case. On the other hand, if

Δ_{q} ↛ 0

, then

A = [0, 1]

and

\sum_{q = 1}^{\infty} φ (q) Δ_{q}^{'} = \infty

, so that Catlin's conjecture is trivially true.

Just like in Theorem 2.3 of JarnÃk and Besicovitch, it would be important to also have information about the Hausdorff dimension of the sets

A

and

A^{*}

in the case when they have null Lebesgue measure. In light of relation (2.6), it suffices to answer this question for the latter set. Beresnevich and Velani [4] proved the remarkable result that the DuffinSchaeffer conjecture implies a Hausdorff measure version of itself. This is a consequence of a much more general Mass Transfer Principle that they established, and which allows transfering information concerning the Lebesgue measure of certain lim sup sets to the Hausdorff measure of rescaled versions of them. As a corollary, they proved:

Theorem 2.6 (Beresnevich-Velani (2006)). Assume that the Duffin-Schaeffer conjecture is true. Let

Δ_{1}, Δ_{2}, \dots ⩾ 0

be such that

\sum_{q = 1}^{\infty} φ (q) Δ_{q} < \infty

. Then the Hausdorff dimension of the set

A^{*}

defined by (2.4) equals the infimum of the set of

s > 0

such that

\sum_{q = 1}^{\infty} φ (q) Δ_{q}^{s} < \infty

2.3. Progress towards the Duffin-Schaeffer conjecture

Since its introduction in 1941, the Duffin-Schaeffer conjectured has been the subject of intensive research activity, with various special cases proven over the years. This process came to a conclusion recently with the proof of the full conjecture [20].

Theorem 2.7 (Koukoulopoulos-Maynard (2020)). The Duffin-Schaeffer conjecture is true.

We will outline the main ideas of the proof of Theorem 2.7 in Section 3. But first we give an account of the work that preceeded it.

Notation. Given two functions

f, g : X \to R

, we write

f (x) ≪ g (x)

(or

f (x) = O (g (x)

)) for all

x \in X

to mean that there is a constant

C

such that

| f (x) | ⩽ C g (x)

for all

x \in Y

In the same paper where they introduced their conjecture, Duffin and Schaeffer proved the first general case of it:

Theorem 2.8 (Duffin-Schaeffer (1941)). The Duffin-Schaeffer conjecture is true for all sequences

{(Δ_{q})}_{q = 1}^{\infty}

such that

\begin{matrix} (2.7) & \underset{Q \to \infty}{lim sup} \frac{\sum_{q ⩽ Q} φ (q) Δ_{q}}{\sum_{q ⩽ Q} q Δ_{q}} > 0 \end{matrix}

To appreciate this result, we must make a few comments about condition (2.7). Note that its left-hand side is the average value of

φ (q) / q

over

q \in [1, Q]

, where

q

is weighted by

w_{q} := q Δ_{q}

. In particular, we may restrict our attention to

q

with

Δ_{q} > 0

. Now, we know

\frac{φ (q)}{q} = \prod_{p ∣ q} (1 - \frac{1}{p})

In particular,

φ (q) / q ⩽ 1

, and the only way this ratio can become much smaller than 1 is if

q

is divisible by lots of small primes. To see this, let us begin by observing that

q

can have at most

\log q / \log 2

prime factors in total. Therefore,

\begin{matrix} (2.8) & \prod_{p ∣ q, p > \log q} (1 - \frac{1}{p}) ⩾ {(1 - \frac{1}{\log q})}^{\log q / \log 2} ⩾ \frac{1}{5} \end{matrix}

for

q

large enough. In addition, we have

\begin{matrix} (2.9) & \prod_{\begin{matrix} (\log q)^{0.01} < p ⩽ \log q \end{matrix}} (1 - \frac{1}{p}) ⩾ \prod_{(\log q)^{0.01} < p ⩽ \log q} (1 - \frac{1}{p}) ⩾ \frac{1}{200} \end{matrix}

for

q

large enough by Mertens' estimate [19, THEOREM 3.4]. Already the above inequalities show that only the primes

⩽ (\log q)^{0.01}

can affect the size of

φ (q) / q

. But more is true:

φ (q) / q

is small only if

q

is divided by many primes

⩽ (\log q)^{0.01}

. Imagine, for example, that

\begin{matrix} (2.10) & # {p ∣ q : e^{j - 1} < p ⩽ e^{j}} ⩽ e^{j} / j^{2} + 1000 \end{matrix}

for

j = 1, 2, \dots, 1 + ⌊ 0.01 \log \log q ⌋

. We would then have

\prod_{\begin{matrix} j - 1 < p ⩽ e^{j} \\ p ∣ q \end{matrix}} (1 - \frac{1}{p}) ⩾ {(1 - \frac{1}{e^{j}})}^{e^{j} / j^{2} + 1000} = \exp (- 1 / j^{2} + O (e^{- j}))

Multiplying this over all

j

, we deduce that

φ (q) / q ⩾ c

for some

c > 0

independent of

q

We have thus proven that for (2.7) to fail, the main contribution to the weighted sum

\sum_{q ⩽ Q} w_{q}

with

w_{q} = q Δ_{q}

must come from integers for which (2.10) fails. As a matter of fact, (2.10) must fail for lots of

j

's. This is an extremely rare event if we choose

q

uniformly at random from

[1, Q]

(or even if we choose it uniformly at random from various "nice" subsets of

[1, Q]

, such as the primes, or the values of a monic polynomial with integer coefficients).

p ⩽ e^{j}}

with respect to the uniform counting measure on

[1, Q]

. We have

\frac{1}{Q} \sum_{q ⩽ Q} # {p ∣ q : e^{j - 1} < p ⩽ e^{j}} = \sum_{e^{j - 1} < p ⩽ e^{j}} \frac{# {q ⩽ Q : p ∣ q}}{Q} ⩽ \sum_{e^{j - 1} < p ⩽ e^{j}} \frac{1}{p} ≪ \frac{1}{j}

by Mertens' theorem [19, THEOREM 3.4]. This is much smaller than

e^{j} / j^{2}

, so (2.10) should fail rarely as

j \to \infty

. (For instance, we may use Markov's inequality to see this claim.)

In conclusion, Theorem 2.8 settles the Duffin-Schaeffer conjecture when

Δ_{q}

is mainly supported on "normal" integers, without too many small prime factors. In particular, it implies a significant improvement of Theorems 1.3 and 1.4 for almost all

x \in R

Corollary 2.9. For almost all

x \in R

, there are infinitely many reduced fractions

a / p

and

b / q^{2}

such that

p

is prime,

| x - a / p | < p^{- 2}

and

| x - b / q^{2} | < q^{- 3}

The next important step towards the Duffin-Schaeffer conjecture is a remarkable zero-one law due to Gallagher [12].

Theorem 2.10 (Gallagher (1961)). If

A^{*}

is as in (2.4), then meas

(A^{*}) \in {0, 1}

Gallagher's theorem says grosso modo that either we chose the quantities

Δ_{q}

to be "too small" and thus missed almost all real numbers, or we chose them "sufficiently large" so that almost all numbers have the desired rational approximations. The Duffin-Schaeffer conjecture is then the simplest possible criterion to decide in which case we are.

The proof of Theorem 2.10 is a clever adaptation of an ergodic-theoretic argument due to Cassels [6] in the simpler setting of nonreduced rational approximations. We give Cassel's proof and refer the interested readers to

[12, 14]

for the proof of Theorem 2.10.

Theorem 2.11 (Cassels (1950)). If

A

is as in (2.1), then meas(A)

) \in {0, 1}

Proof. We need the following fact [14, LEMMA 2.1] that uses Lebesgue's Density Theorem: Let

I_{1}, I_{2}, \dots, J_{1}, J_{2}, \dots

be intervals of lengths tending to 0 , and let

c > 0

. For all

k

, suppose

J_{k} \subseteq I_{k}

and meas

(J_{k}) ⩾ c

meas

(I_{k})

. Then meas(lim sup

_{k \to \infty} I_{k} ∖ lim sup sup_{k \to \infty} J_{k}) = 0

Now, for each

r ⩾ 1

, let

A^{(r)}

be defined as in (2.1) but with

Δ_{q} / r

in place of

Δ_{q}

. Hence, meas

(A ∖ A^{(r)}) = 0

by the above fact. Therefore, if

A^{(\infty)} := ⋂_{n = 1}^{\infty} A^{(n)}

, then meas

(A ∖ A^{(\infty)}) = 0

. Now, consider the map

ψ : [0, 1] \to [0, 1]

defined by

ψ (x) := {2 x}

, and note that

ψ (A^{(\infty)}) \subseteq A^{(\infty)}

. In particular,

\frac{1}{N} \sum_{n = 0}^{N - 1} 1_{A^{(\infty)}} (ψ^{n} (x)) = 1

for all

x \in A^{(\infty)}

and all

N \in N

. Since

ψ

is ergodic with respect to the Lebesgue measure [26, P. 293 â‚¹ 305-6], Birkhoff's Ergodic Theorem [26, ÑH. 6, coR. 5.6] implies that meas

(A^{(\infty)}) \in {0, 1}

The first significant step towards establishing the Duffin-Schaeffer conjecture for irregular sequences

Δ_{q}

, potentially supported on integers with lots of small prime factors, was carried out by ErdÅ‘s [10] and Vaaler [27].

Theorem 2.12 (ErdÅ‘s (1970), Vaaler (1978)). The Duffin-Schaeffer conjecture is true for all sequences

{(Δ_{q})}_{q = 1}^{\infty}

such that

Δ_{q} = O (1 / q^{2})

for all

q

This theorem is, of course, most interesting when

\sum_{q = 1}^{\infty} φ (q) Δ_{q} = \infty

. Since

Δ_{q} = O (1 / q^{2})

and

φ (q) ⩽ q

, we find

\sum_{q \in S} 1 / q = \infty

with

S = {q : Δ_{q} > 0}

. In particular,

S

must be somewhat dense in

N

. Therefore, Theorem 2.12 has the advantage over Theorem 2.8 that

S

can contain many irregular integers, and the disadvantage that it has to be quite dense.

The Duffin-Schaeffer conjecture has a natural analogue in

R^{k}

with

k ⩾ 2

: given

Δ_{1}, Δ_{2}, \dots ⩾ 0

, let

A^{*} (k)

be the set of

\vec{x} = (x_{1}, \dots, x_{k}) \in R^{k}

for which there are infinitely many

k

-tuples

(a_{1} / q, \dots, a_{k} / q)

of reduced fractions with

| x_{j} - a_{j} / q | < Δ_{q}

for all

j

. Then

A^{*} (k)

should contain almost no or almost all

\vec{x} \in R^{k}

, according to whether the series

\sum_{q = 1}^{\infty} {(φ (q) Δ_{q})}^{k}

converges or diverges. This was proven by Pollington and Vaughan [22].

Theorem 2.13 (Pollington-Vaughan (1990)). The

k

-dimensional Duffin-Schaeffer conjecture is true for all

k ⩾ 2

Following this result, a lot of research focused on proving the Duffin-Schaeffer conjecture when the series

\sum_{q = 1}^{\infty} φ (q) Δ_{q}

diverges fast enough (see, e.g., [14, THEOREM 3.7(III)], [3,15]). Aistleitner, Lachmann, Munsch, Technau, and Zafeiropoulos [2] proved the Duffin-

Schaeffer conjecture when

\sum_{q = 1}^{\infty} φ (q) Δ_{q} / (\log q)^{ε} = \infty

for some

ε > 0

. A report by Aistleitner [1], announced at the same time as [20], explains how to replace

(\log q)^{ε}

(\log \log q)^{ε}

3. THE MAIN INGREDIENTS OF THE PROOF OF THE DUFFIN-SCHAEFFER CONJECTURE

3.1. Borel-Cantelli without independence

Recall the definition of the sets

A_{q}^{*}

in (2.5). Let us assume that

Δ_{q} ⩽ 1 / (2 q)

for all

q

(cf. (2.3)) so that meas

(A_{q}^{*}) = 2 φ (q) Δ_{q} \in [0, 1]

, and let us also suppose that

\sum_{q = 1}^{\infty} φ (q) Δ_{q} = \infty

. The first technical difficulty we must deal with is how to prove an analogue of the second Borel-Cantelli lemma (cf. Theorem 2.4(b)) without assuming that the events

A_{q}^{*}

are independent. We follow an idea due to Turan, which is already present in

[8]

By Gallagher's zero-one law, it is enough to show that meas

(A^{*}) > 0

. Since

⋃_{q ⩾ Q} A_{q}^{*} ↘ A^{*}

, we may equivalently prove that there is some constant

c > 0

such that meas

(⋃_{q ⩾ Q} A_{q}^{*}) ⩾ c

for all large

Q

. In order to limit the potential overlap among the sets

A_{q}^{*}

, we only consider an appropriate subset of them. Since meas

(A_{q}^{*}) = 2 φ (q) Δ_{q} \in [0, 1]

for all

q

, and since

\sum_{q ⩾ Q} φ (q) Δ_{q} = \infty

, there must exist some

R ⩾ Q

such that

\begin{matrix} (3.1) & 1 ⩽ \sum_{q \in [Q, R]} meas (A_{q}^{*}) ⩽ 2 \end{matrix}

We will only use the events

A_{q}^{*}

with

q \in [Q, R]

. We trivially have the union bound

meas (⋃_{q \in [Q, R]} A_{q}^{*}) ⩽ \sum_{q \in [Q, R]} meas (A_{q}^{*}) ⩽ 2

If we can show that the sets

A_{q}^{*}

with

q \in [Q, R]

do not overlap too much, so that

\begin{matrix} (3.2) & meas (⋃_{q \in [Q, R]} A_{q}^{*}) ⩾ c \sum_{q \in [Q, R]} meas (A_{q}^{*}) ⩾ c \end{matrix}

we will be able to deduce that

meas (⋃_{q \in [Q, R]} A_{q}^{*}) ⩾ c

and a fortiori that meas

(⋃_{q ⩾ Q} A_{q}^{*}) ⩾ c

. As the following lemma shows, (3.2) is true under (3.1) as long as we can control the correlations of the events

A_{q}^{*}

on average.

Lemma 3.1. Let

E_{1}, \dots, E_{k}

be events in the probability space

(Ω, F, P)

. We then have that

P (⋃_{j = 1}^{k} E_{j}) ⩾ \frac{{(\sum_{j = 1}^{k} P (E_{j}))}^{2}}{\sum_{i, j = 1}^{k} P (E_{i} \cap E_{j})}

Proof. Let

N = \sum_{j} 1_{E_{j}}

. We then have

E [N] = \sum_{j} P (E_{j})

. On the other hand, the CauchySchwarz inequality implies that

E [N]^{2} = E {[1_{N > 0} \cdot N]}^{2} ⩽ P (supp (N)) \cdot E [N^{2}]

Since

supp (N) = ⋃_{j} E_{j}

and

N^{2} = \sum_{i, j} 1_{E_{i} \cap E_{j}}

, the lemma follows.

The following proposition summarizes the discussion of this section.

Proposition 3.2. Let

Δ_{1}, Δ_{2}, \dots ⩾ 0

, and let

A_{q}^{*}

be as in (2.5).

(a) If

C > 0

and

R ⩾ Q ⩾ 1

are such that

\begin{matrix} (3.3) & 1 ⩽ \sum_{q \in [Q, R]} meas (A_{q}^{*}) ⩽ 2 and \sum_{Q ⩽ q < r ⩽ R} meas (A_{q}^{*} \cap A_{r}^{*}) ⩽ C \end{matrix}

then

meas (⋃_{q \in [Q, R]} A_{q}^{*}) ⩾ 1 / (2 + 2 C)

(b) If there are infinitely many disjoint intervals

[Q, R]

satisfying (3.3) with the same constant

C > 0

, then meas(

lim sup_{q \to \infty} A_{q}^{*}) = 1

3.2. A bound on the pairwise correlations

As per Proposition 3.2, we need to control the correlations of the events

A_{q}^{*}

. To this end, we have a lemma of Pollington-Vaughan [22] (see also [10, 27]).

Lemma 3.3. Let

q, r

be two distinct integers

⩾ 2

, let

Δ_{q}, Δ_{r} ⩾ 0

, let

A_{q}^{*}, A_{r}^{*}

be as in (2.5), and let

M (q, r) = 2 max {Δ_{q}, Δ_{r}} lcm [q, r]

. If

M (q, r) ⩽ 1

, then

A_{q}^{*} \cap A_{r}^{*} = \emptyset

. Otherwise,

meas (A_{q}^{*} \cap A_{r}^{*}) ≪ φ (q) Δ_{q} \cdot φ (r) Δ_{r} \cdot \exp (\sum_{\begin{matrix} p ∣ q r / \gcd (q, r) \\ p > M (q, r) \end{matrix}} \frac{1}{p})

Proof. Let

Δ = max {Δ_{q}, Δ_{r}}, δ = min {Δ_{q}, Δ_{r}}

and

M = M (q, r)

. The intervals

I_{a} = (\frac{a}{q} - Δ_{q}, \frac{a}{q} + Δ_{q})

and

J_{b} = (\frac{b}{r} - Δ_{r}, \frac{b}{r} + Δ_{r})

intersect only if

2 Δ > | \frac{a}{q} - \frac{b}{r} |

. Since the right-hand side is

⩾ 1 / lcm [q, r]

when

\gcd (a, q) = \gcd (b, r) = 1

, we infer that

A_{q}^{*} \cap A_{r}^{*} = \emptyset

M ⩽ 1

Now, assume that

M > 1

. Since meas

(I_{a} \cap J_{b}) ⩽ 2 δ

for all

a, b

, we have

meas (A_{q}^{*} \cap A_{r}^{*}) ⩽ 2 δ \cdot # {\begin{cases} 1 ⩽ a ⩽ q, & \gcd (a, q) = 1 \\ 1 ⩽ b ⩽ r, & \gcd (b, r) = 1 \end{cases} : | \frac{a}{q} - \frac{b}{r} | < 2 Δ}

Let

a / q - b / r = m / lcm [q, r]

. Then

1 ⩽ | m | ⩽ M

and

\gcd (m, q_{1} r_{1}) = 1

, where

q_{1} = q / d

and

r_{1} = r / d

with

d = \gcd (q, r)

3. VOJTA'S APPROACH TO THE MORDELL CONJECTURE

4. COMPARING WEIL AND NÃ‰RON-TATE HEIGHTS

4.1. Degenerate subvarieties and the Betti map

4.2. Comparing the Weil and NÃ©ron-Tate heights on a subvariety

5. APPLICATION TO MODERATE POINTS ON CURVES

5.1. The Faltings-Zhang morphism

5.2. Bounding the number of moderate points-a sketch

6. HYPERELLIPTIC CURVES

ACKNOWLEDGMENTS

FUNDING

REFERENCES

PHILIPP HABEGGER

THETA LIFTING AND LANGLANDS FUNCTORIALITY

ATSUSHI ICHINO

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

2. THETA LIFTING

2.1. Basic definitions and properties

2.2. Explicit realization of the Jacquet-Langlands correspondence

2.3. Seesaw identities

3. THE SHIMURA-WALDSPURGER CORRESPONDENCE

3.1. Modular forms of half-integral weight

3.2. Global correspondence

3.3. Local correspondence

4. GEOMETRIC REALIZATION OF THE JACOUET-LANGLANDS CORRESPONDENCE

ACKNOWLEDGMENTS

FUNDING

REFERENCES

ATSUSHI ICHINO

RATIONAL

APPROXIMATIONS

DIMITRIS KOUKOULOPOULOS

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. DIOPHANTINE APPROXIMATION

1.1. First principles

1.2. Improving Dirichlet's approximation theorem

2. METRIC DIOPHANTINE APPROXIMATION

2.1. The theorems of Khinchin and JarnÃ­k-Besicovitch

2.2. Generalizing Khinchin's theorem

2.3. Progress towards the Duffin-Schaeffer conjecture

3. THE MAIN INGREDIENTS OF THE PROOF OF THE DUFFIN-SCHAEFFER CONJECTURE

3.1. Borel-Cantelli without independence

3.2. A bound on the pairwise correlations

2.1. The theorems of Khinchin and JarnÃk-Besicovitch